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WinBUGS User Manual - Department of Mathematics and Statistics

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1. pD this is the effective number of parameters and is given by pD Dbar Dhat Thus pD is the posterior mean of the deviance minus the deviance of the posterior means DIC this is the Deviance Information Criterion and is given by DIC Dbar pD Dhat 2 pD The model with the smallest DIC is estimated to be the model that would best predict a replicate dataset of the same structure as that currently observed e e The Info Menu BUGS Contents General properties Open Log Clear Log Node info Components General properties top home This menu allows the user to obtain more information about how the software is working Open Log top home This option opens a log window to which error and status information is written Clear Log top home This option clears all the information displayed in the log window Node info top home node values methods node state This dialog box allows information about particular nodes in the model to be obtained node the name of the node for which information is required should be typed here values displays the current value of the node for each chain in the log window methods displays the type of updater used to sample from the node if appropriate in the log window node displays the type used to represent the node in the log window state opens a trap window showing the current state of the internal data structures
2. 20 file array dat c 1 2 3 4 5 Dim c 2 5 The file is now in the correct 6 7 8 9 10 11 12 USCH 21 22 23 24 Dim c 4 2 3 Edit the Dim statement in this file from correct format to input the required 3 2 4 dimensional array into WinBUGS in the order L j k th element of matrix 1 1 1 1 1 2 Dim c 4 2 3 to value 2 Dim c 3 2 4 The file is now in the 1 2 4 5 11 2 2 e 2 1 3 11 2 1 4 12 2 2 4 13 2 2 2 14 ae BR 2 A 24 Rectangular format The columns for data in rectangular format need to be headed by the array name The arrays need to be of equal size and the array names must have explicit brackets for example age sex 26 0 52 1 34 0 END Note that the file must end with an END statement as shown above and below and this must be followed by at least one blank line Multi dimensional arrays can be specified by explicit indexing for example the Ratsy file begins Y 1 Y 2 Y 3 Y 4 DER 15 199 246 283 320 145 199 249 293 354 147 214 263 312 328 153 200 244 286 324 END The first index position for any array must always be empty It is possible to load a mixture of rectangular and S Plus format data files for the same model For example if data arrays are provided in a rectangular file constants can be defined in a separate list statement see also the Rats example with data f
3. 14 1 Person 3 11 0 9 7 10 3 9 6 There are three different ways of entering such ragged data into WinBUGS 1 Fill to rectangular Here the data is padded out by explicitly including the missing data i e 13 2 NA NA NA 122 8 14 1 NA NA 11 0 927 10 3 9 56 END or list y structure Data c 13 2 NA NA NA 12 3 14 1 NA NA 11 0 9 7 10 3 9 6 Dim c 3 4 A model suchas y i j dnorm mu i 1 can then be fitted This approach is inefficient unless one explicitly wishes to estimate the missing data 2 Nested indexing Here the data are stored in a single array and the associated person is recorded as a factor i e yt person 13 2 1 12 3 2 14 1 2 11 0 3 9 7 3 10 3 3 9 6 3 END or list y c 13 2 12 3 14 1 11 0 9 7 10 3 9 6 person c 1 2 2 3 3 3 3 A model such as y k dnorm mu person k 1 can then ke fitted This seems an efficient and clear way to handle the problem 3 Offset Here an offset array holds the position in the data array at which each person s data starts For example the data might be list y c 13 2 12 3 14 1 11 0 9 7 10 3 9 6 offset c 1 2 4 8 and lead to a model containing the code for k in offset i offset i 1 1 ylk dnorm mu i 1 The danger with this method is that it relies on getting the offsets correct and they are difficult to check See the ragged example for a full worked example of these
4. Volume 1 and Volume 2 An alternative way of specifying a model in WinBUGS is to use the graphical interface known as DoodleBUGS Running a model in WinBUGS top rome The WinBUGS software uses compound documents which comprise various different types of information formatted text tables formulae plots graphs etc displayed in a single window and stored in a single file This means that it is possible to run the model for the seeds example directly from this tutorial document since the model code can be made live just by highlighting it However it is more usual when creating your own models to have the model code and data etc in separate files We have therefore created a separate file with the model code in it for this tutorial you will find the file in Manuals Tutorial seeds_model odc or click here Step 1 Open the seeds_model file as follows Point to File on the tool bar and click once with the left mouse button LMB Highlight the Open option and click once with LMB Select the appropriate directory and double click on the file to open Step 2 To run the model we first need to check that the model description does fully define a probability model Point to Model on the tool bar and click once with LMB Highlight the Specification option and click once with LMB Focus the window containing the model code by clicking the LMB once anywhere in the window the top panel of the window should then
5. dim p Y p i 1 r 1 p 7 x 0 1 2 Ae Pe T ean r E r top home T b wein pjt E 4 D 0 lt p lt 1 D a P d x dchisqr k Double Exponential x ddexp mu tau Exponential x dexp Lambda Gamma x dgamma r mu Generalized Gamma x gen gamma r mu beta Log normal x dlnorm mu tau Logistic x dlogis mu tau Normal x dnorm mu tau Pareto x dpar alpha c Student t x dt mu tau k Uniform x dunif a b 9 k 2pk 2 1 7 2 H Aar S z exp 7 HIR co lt 2 lt co Ae a2 gt 0 ryt e7Ht H E zU I r p u r exp ux 2 gt 0 I r a exp 3 logx u gt 0 2T x 2 Texp T x ul 1 exp t x p oo lt T lt 00 actg a AKT Jerta e S IER r 1 Zo D Weibull x dweib v lambda vAxr texp Az gt 0 Discrete Multivariate top home Multinomial x dmulti p N 23 D L S IL l l Ti Continuous Multivariate top home Dirichlet pl ddirch alpha T i t Tpm IL I as In Dn pi l Multivariate Normal x dmnorm mu T 2r 4 2 T 2exp 4 x py T ul oo lt T lt 00 Multivariate Student t x dmt mu T k T k d 2 T k 2 Lidd x 1 t x p T x ui E oo KS 2 re k d 2 Wishart x dwish R k R 2 amp P 1 2 exp 4 Tr R2
6. node defines the vector of nodes to be compared with each other As the comparisons are with respect to posterior distributions node must be a monitored variable other where appropriate other defines a vector of reference points to be plotted alongside each element of node on the same scale for example other may be the observed data in a model fit plot see below The elements of other may be either monitored variables in which case the posterior mean is plotted or they may be observed known axis where appropriate axis defines a set of values against which the elements of node and other should be plotted Each element of axis must be either Known observed or alternatively a monitored variable in which case the posterior mean is used Note node other and axis should all have the same number of elements beg and end are used to select the subset of stored samples from which the desired plot should be derived box plot this command button produces a single plot in which the posterior distributions of all elements of node are summarised side by side For example box plot beta By default the distributions are plotted in order of the corresponding variable s index in node and are also labelled with that index Boxes represent inter quartile ranges and the solid black line at the approximate centre of each box is the mean the arms of each box extend to cover the central 95 per cent of the distribution thei
7. beta0 betal zl1l i beta2 z2 1i b i The following functions can be used on the left hand side of logical nodes as link functions log Logit cloglog and probit where probit x lt yis equivalentto x lt phi y It is important to keep in mind that logical nodes are included only for notational convenience they cannot be given data or initial values except when using the data transformation facility described below Deviance A logical node called deviance is created automatically by WinBUGS this stores 2 log likelinood where likelinood is the conditional probability of all data nodes given their stochastic parent nodes This node can be monitored and contributes to the DIC function see DIC Arrays and indexing top home Arrays are indexed by terms within square brackets The four basic operators and along with appropriate bracketing are allowed to calculate an integer function as an index for example YL J Re A On the left hand side of a relation an expression that always evaluates to a fixed value is allowed for an index whether it is a constant or a function of data On the right hand side the index can be a fixed value or a named node which allows a straightforward formulation for mixture models in which the appropriate element of an array is picked according to a random quantity see Nested indexing and mixtures However functions of unobserved nodes are not permi
8. the seeds example set monitors for the parameters alphaO alphal alpha2 alphal2 and sigma see here for details on how to do this To run the simulation Select the Update option from the Model menu Type the number of updates iterations of the simulation you require in the appropriate white box labelled updates the default value is 1000 Click once on the update button the program will now start simulating values for each parameter in the model This may take a few seconds the box marked iteration will tell you how many updates have currently been completed The number of times this value is revised depends on the value you have set for the refresh option in the white box above the iteration box The default is every 100 iterations but you can ask the program to report more frequently by changing refresh to say 10 or 1 A sensible choice will depend on how quickly the program runs For the seeds example experiment with changing the refresh option from 100 to 10 and then 1 When the updates are finished the message updates took s will appear in the bottom left of the WinBUGS program window where is the number of seconds taken to complete the simulation If you previously set monitors for any parameters you can now check convergence and view graphical and numerical summaries of the samples Do this now for the parameters you monitored in the seeds example see Checking convergence for tips on ho
9. wj ph xj where is the smoothing parameter defined in the Scatterplot properties dialogue box next to the smooth radio button The default setting for is somewhat arbitrarily max x min x 20 8 Tips and Troubleshooting lA BUGS Contents Restrictions when modelling Some error messages Some Trap messages The program hangs Speeding up sampling Improving convergence This section covers a range of problems that might arise 1 restrictions in setting up the model 2 error messages generated when setting up the model or when sampling 3 Traps these corresponds to an error which has not been picked up by WinBUGS It is difficult to interpret the Trap message 4 the program just hanging 5 slow sampling 6 lack of convergence Restrictions when modelling top home Restrictions have been stated throughout this manual A summary list is as follows a Each stochastic and logical node must appear once and only once on the left hand side of an expression The only exception is when carrying out a data transformation see Data transformations This means for example that it is generally not possible to give a distribution to a quantity that is specified as a logical function of an unknown parameter b Truncated sampling distributions cannot be handled using the I construct see Censoring and truncation c Multivariate distributions Dirichlet and Wishart distributions may only
10. 1 If the file stem is left blank the CODA output is written to windows one for each chain plus one for the index If the file stem is not blank then the CODA output is written to separate text files each starting with file stem 2 If the file ends with txt the log window is saved to a text file with all graphics fonts etc stripped out otherwise the log window is saved as is 3 Runs the script stored in file Tricks Advanced Use of the BUGS Language Contents Specifying a new sampling distribution Specifying a new prior distribution Specifying a discrete prior on a set of values Using pD and DIC Mixtures of models of different complexity Where the size of a set is a random quantity Assessing sensitivity to prior assumptions Modelling unknown denominators Handling unbalanced datasets Learning about the parameters of a Dirichlet distribution Use of the cut function Specifying a new sampling distribution top home Suppose we wish to use a sampling distribution that is not included in the list of standard distributions in which an observation x i contributes a likelihood term L i We may use the zeros trick a Poisson phi observation of zero has likelihood exp phi so if our observed data is a set of 0 s and phi i is set to log LIi we will obtain the correct likelihood contribution Note that phi i should always be gt 0 as it is a Poisson mean and so we may need to add a suitable constant to ensure that
11. The load inits button acts on the focus view it will be greyed out unless the focus view contains text The initial values will be loaded for the chain indicated in the text entry field to the right of the caption for chain The value of this text field can be edited to load initial values for any of the chains Initial values are specified in exactly the same way as data files If some of the elements in an array are known say because they are constraints in a parameterisation those elements should be specified as missing NA in the initial values file This command becomes active once the model has been successfully compiled and checks that initial values are in the form of an S Plus object or rectangular array and that they are consistent with any previously loaded data Any syntax errors or inconsistencies in the initial value are displayed If after loading the initial values the model is fully initialized this will be reported by displaying the message initial values loaded model initialized Otherwise the status line will show the message initial values loaded model contains uninitialized nodes The second message can have several meanings a If only one chain is simulated it means that the chain contains some nodes that have not been initialized yet b If several chains are to be simulated it could mean that no initial values have been loaded for one of the chains In either case further initial values can be loaded or the gen
12. an existing Doodle e g from Examples Volume 1 perhaps to fit a problem of your own 5 Try constructing a Doodle from scratch Note that there are many features in the BUGS language that cannot be expressed with Doodles If you wish to proceed to serious non educational use you may want to dispense with DoodleBUGS entirely or just use it for initially setting up a simplified model that can be elaborated later using the BUGS language Unfortunately we do not have a program to back translate from a text based model description to a Doodle MCMC methods top home Users should already be aware of the background to Bayesian Markov chain Monte Carlo methods see for example Gilks et al 1996 Having specified the model as a full joint distribution on all quantities whether parameters or observables we wish to sample values of the unknown parameters from their conditional posterior distribution given those stochastic nodes that have been observed The basic idea behind the Gibbs sampling algorithm is to successively sample from the conditional distribution of each node given all the others in the graph these are known as full conditional distributions the Metropolis within Gibbs algorithm is appropriate for difficult full conditional distributions and does not necessarily generate a new value at each iteration It can be shown that under broad conditions this process eventually provides samples from the joint posterior distribution of the un
13. and display the ranks of the simulated values in an array node the variable to be ranked must be typed in this text field must be an array set starts building running histograms to represent the rank of each component of node An amount of storage proportional to the square of the number of components of node is allocated Even when node has thousands of components this can require less storage than calculating the ranks explicitly in the model specification and storing their samples and it is also much quicker stats summarises the distribution of the ranks of each component of the variable node The quantiles highlighted in the percentile selection box are displayed histogram displays the empirical distribution of the simulated rank of each component of the variable node clear removes the running histograms for node DIC top home BZ Tool x set clear DIE The DIC Tool dialog box is used to evaluate the Deviance Information Criterion DIC Spiegelhalter et al 2002 and related statistics these can be used to assess model complexity and compare different models Most of the examples packaged with WinBUGS contain an example of their usage It is important to note that DIC assumes the posterior mean to be a good estimate of the stochastic parameters If this is not so say because of extreme skewness or even bimodality then DIC may not be appropriate There are also circumstances such as with mixture model
14. be used as conjugate priors and must have known parameters although there does exist a trick for allowing unknown Dirichlet parameters in contrast multinomial distributions may not be used as priors and the order N of the distribution must be known multivariate normal and Student t distributions can be used anywhere in the graph i e as prior or likelihood and there are no restrictions regarding their parameters See Constraints on using certain distributions d Logical nodes cannot be given data or initial values This means for example that it is not possible to model observed data that is the sum of two random variables See Logical nodes e Poisson priors can only be specified for the unknown order n of a single binomial observation In this case the sampling method for a shifted Poisson distribution will be used see MCMC methods If more than one binomial response is assumed to share the same order parameter then the algorithm for a shifted Poisson cannot be used to update the order parameter and WinBUGS will return the error message Unable to choose update method for n A suitably constrained continuous prior may be used for the order parameter provided that it is used in conjunction with the round function to ensure integer values see here for an example of using the round function to this end Some error messages top home a expected variable name indicates an inappropriate variable name b lin
15. dereference read can occur at compilation in some circumstances when an inappropriate transformation is made for example an array into a scalar e Trap messages referring to DFreeARS indicate numerical problems with the derivative free adaptive rejection algorithm used for log concave distributions One possibility is to change to Slice sampling see here for details The program hangs top home This could be due to a a problem that seems to happen with NT rebooting is a crude way out b interference with other programs running try to run WinBUGS on its own c a particularly ill posed model try the approaches listed above under Trap messages Speeding up sampling top home The key is to reduce function evaluations by expressing the model in as concise a form as possible For example take advantage of the functions provided and use nested indexing wherever possible Look at the packaged examples and others provided on users web sites Improving convergence top home Possible solutions include a better parameterisation to improve orthogonality of joint posterior b standardisation of covariates to have mean 0 and standard deviation 1 c use of ordered over relaxation For other problems try the FAQ pages of the web site and the Classic BUGS manual Then try mailing us on bugs mrc bsu cam ac uk Please try and keep the discussion list for modelling issues and problems other than my program wont r
16. functions in Table below can also be used in logical expressions In Table function arguments represented by e can be expressions those by s must be scalar valued nodes in the graph and those represented by v must be vector valued nodes in a graph Table I Functions abs e e cloglog e In In 1 e cos e cosine e cut e cuts edges in the graph see Use of the cut function equals el e2 1 if e1 e2 0 otherwise exp e exp e inprod vl v2 Live interp lin e vl v2 V2p D n 44 7 V2p Miel Ving 4 7 Mp where the elements of vi are in ascending order and p is such that vip lt lt Men inverse v v1 for symmetric positive definite matrix v log e In e logdet v In det v for symmetric positive definite v logfact e Ine loggam e In T e logit e In e 1 e max el e2 e1 if e1 gt e2 ei otherwise mean v my n dim v min el e2 el if e1 lt e2 e2 otherwise phi e standard normal cdf pow el e2 e12 sin e sine e sqrt e ell rank v s number of components of v less than or equal to vs ranked v s the st smallest component of v round e nearest integer to e sd v standard deviation of components of v n 1 in denominator step e 1 if e gt 0 0 otherwise sum v Dy trunc e greatest integer less than or equal to e A link function can also be specified acting on the left hand side of a logical node e g logit mu i lt
17. incoming edge is to be deleted and then click into its parent while holding down the ctrl key Moving a Doodle top home After constructing a Doodle it can be moved into a document that may also contain data initial values and other text and graphics This can be done by choosing Select Document from the Edit menu and then either copying and pasting or dragging the Doodle Resizing a Doodle top home To change the size of Doodle which is already in a document containing text click once into the Doodle with the left mouse button A narrow border with small solid squares at the corners and mid sides will appear Drag one of these squares with the mouse until the Doodle is of the required size Printing a Doodle top home H a Doodle is in its own document it may be printed directly from the File menu If a postscript version of a Doodle is required you could install a driver for a postscript printer say Apple LaserWriter but set it up to print to file checking the paper size is appropriate Alternatively Doodles can be copied to a presentation or word processing package and printed from there The Model Menu BUGS Contents General properties Specification Update Monitor Metropolis Save State Seed Script General properties top home The commands in this menu either apply to the whole statistical model or open dialog boxes This menu is only on display if the focus view is a text window or
18. indicates numerical overflow Possible reasons include initial values generated from a vague prior distribution may be numerically extreme specify appropriate initial values numerically impossible values such as log of a non positive number check for example that no zero expectations have been given when Poisson modelling numerical difficulties in sampling Possible solutions include better initial values more informative priors uniform priors might still be used but with their range restricted to plausible values better parameterisation to improve orthogonality standardisation of covariates to have mean 0 and standard deviation 1 can happen if all initial values are equal Probit models are particularly susceptible to this problem i e generating undefined real results If a probit is a stochastic node it may help to put reasonable bounds on its distribution e g probit p i lt delta il delta i dnorm mu i tau I 5 5 This trap can sometimes be escaped from by simply clicking on the update button The equivalent construction p i lt phi delta i may be more forgiving b index array out of range possible reasons include attempting to assign values beyond the declared length of an array if a logical expression is too long to evaluate break it down into smaller components c stack overflow can occur if there is a recursive definition of a logical node d NIL
19. inits button can be pressed to try and generate initial values for all the uninitialized nodes in all the simulated chains Generally it is recommended to load initial values for all fixed effect nodes founder nodes with no parents for all chains initial values for random effects can be generated using the gen inits button This load inits button can still be executed once Gibbs sampling has been started It will have the effect of starting the sampler out on a new trajectory A modal warning message will appear if the command is used in this context gen inits The gen inits button attempts to generate initial values by sampling either from the prior or from an approximation to the prior In the case of discrete variables a check is made that a configuration of zero probability is not generated This command will generate extreme values if any of the priors are very vague If the command is successful the message initial values generated model initialized is displayed otherwise the message could not generate initial values is displayed The gen inits button becomes active once the model has been successfully compiled and will cease to be active once the model has been initialized Update top home PEN updates ooo refresh fi oo update thin fi iteration jo J over relax IV adapting This command will become active once the model has been compiled and initialized and has fields updates number of MCMC upd
20. sets are needed for this tutorial since we have specified two chains these are stored in file Manuals Tutorial seeds_inits odc Open this file now To load the initial values Highlight the word 1 ist at the beginning of the first set of initial values Click once with the LMB on the load inits button in the Specification Tool window A message saying initial values loaded model contains uninitialized nodes try running gen inits or loading more files should appear in the bottom left of the WinBUGS program window Repeat this process for the second initial values file A message saying initial values loaded model initialized should now appear in the bottom left of the WinBUGS program window Note that you do not need to provide a list of initial values for every parameter in your model You can get WinBUGS to generate initial values for any stochastic parameter not already initialized by clicking with the LMB on the gen inits button in the Specification Tool window WinBUGS generates initial values by forward sampling from the prior distribution for each parameter Therefore you are advised to provide your own initial values for parameters with vague prior distributions to avoid wildly inappropriate values Step 7 Close the Specification Tool window You are now ready to start running the simulation However before doing so you will probably want to set some monitors to store the sampled values for selected parameters For
21. show baseline this check box should be used to specify whether or not a baseline should be shown on the plot the numeric field to the immediate right of the check box gives the value of that baseline The default setting is that a baseline equal to the global mean of the posterior means will be shown the user may specify an alternative baseline simply by editing the displayed value show labels check box that determines whether or not each distribution box should be labelled with its index in node that is node on the Comparison Tool The default setting is that labels should be shown show means or show medians these radio buttons specify whether the solid black line at the approximate centre of each box is to represent the posterior mean or the posterior median mean is the default rank use this check box to specify whether the distributions should be ranked and plotted in order The basis for ranking is either the posterior mean or the posterior median depending on which is chosen to be displayed in the plot via show means or show medians vert box or horiz box these radio buttons determine the orientation of the plot The default is vertical boxes which means that the scale axis i e that which measures the width of the distributions is the y axis log scale the scale axis can be given a logarithmic scale by checking this check box Finally in the bottom right hand corner there is a colour field for selecting the f
22. the use of DIC and pD for a full discussion see Spiegelhalter et al 2002 1 DIC is intended as a generalisation of Akaike s Information Criterion AIC For non hierarchical models pD should be approximately the true number of parameters 2 Slightly different values of Dhat and hence pD and DIC can be obtained depending on the parameterisation used for the prior distribution For example consider the precision tau 1 variance of a normal distribution The two priors tau dgamma 0 001 0 001 and log tau dunif 10 10 log tau lt log tau are essentially identical but will give slightly different results for Dhat for the first prior the stochastic parent is tau and hence the posterior mean of tau is substituted in Dhat while in the second parameterisation the stochastic parent is og tau and hence the posterior mean of log tau is substituted in Dhat 3 For sampling distributions that are log concave in their stochastic parents pD is guaranteed to be positive provided the simulation has converged However it is theoretically possible to get negative values We have obtained negative pD s in the following situations i with non log concave likelihoods e g Student t distributions when there is substantial conflict between prior and data ii when the posterior distribution for a parameter is symmetric and bimodal and so the posterior mean is a very poor summary statistic and gives a very large deviance
23. using beta g j in a regression equation In the BUGS language nested indexing can be used for the parameters of distributions for example the Eyes example concerns a normal mixture in which the ith case is in an unknown group T which determines the mean r of the measurement y Hence the model is T Categorical P y Normal A T which may be written in the BUGS language as for i in 1 N Tie dear de y i dnorm lambda T i tau However when using Doodles the parameters of a distribution must be a node in the graph and so an additional stage is needed to specify the mean A Ar as shown in the graph below We emphasise the care required in establishing convergence of these notorious models name mu i type logical link identity value lambda T i lambda k for k IN 1 2 for i IN 1 N Vector parameters can also be identified dynamically but currently only to a maximum of two dimensions For example if we wanted a two state categorical variable x to have a vector of probabilities indexed by i and j then we could write x dcat p i jJ 1 2 However suppose we require three level indexing for example a deat p a 1 2 b deat p b 1 2 deat p c 1 2 d deat p dla D cc 172 WinBUGS will not permit this and so the index must be explicitly calculated d deat p k 1 2 k lt 8 a
24. values for many variables We recommend setting summary monitors on long vectors of parameters such as random effects in order to store posterior summaries and then also setting full samples monitors on a small subset of the random effects plus other relevant parameters e g means and variances to check convergence To set a summary monitor Select Summary from the Inference menu Type the name of the parameter to be monitored in the white box marked node Click once with the LMB on the button marked set Repeat for each parameter to be monitored Note you should not set a summary monitor until you are happy that convergence has been reached see Checking convergence since it is not possible to discard any of the pre convergence burn in values from the summary once it is set other than to clear the monitor and re set it Checking convergence ron home Checking convergence requires considerable care It is very difficult to say conclusively that a chain simulation has converged only to diagnose when it definitely hasn t The following are practical guidelines for assessing convergence For models with many parameters it is impractical to check convergence for every parameter so just choose a random selection of relevant parameters to monitor For example rather than checking convergence for every element of a vector of random effects just choose a random subset say the first 5 or 10 Examine
25. 1 AA b 1 e This calculated index trick is useful in many circumstances Formatting of data top home Data can be S Plus format see most of the examples or for data in arrays in rectangular format The whole array must be specified in the file it is not possible just to specify selected components Missing values are represented as NA All variables in a data file must be defined in a model even if just left unattached to the rest of the model In Doodles such variables can be left as constants in a model description they can be assigned vague priors or allocated to dummy variables S Plus format This allows scalars and arrays to be named and given values in a single structure headed by key word 1 ist There must be no space after list For example in the Rats example we need to specify a scalar xbar dimensions Nand T a vector x and a two dimensional array Y with 30 rows and 5 columns This is achieved using the following format Jett xbar 22 N 30 T 5 x c 8 0 15 0 22 0 29 0 36 0 Y structure Data c 151 199 246 283 320 145 199 249 293 354 137 180 219 258 291 153 200 244 286 324 Dit 06030 5 See the examples for other use of this format WinBUGS reads data into an array by filling the right most index first whereas the S Plus program fills the left most index first Hence WinBUGS reads the string of numbers c 1 2 3 4 5 6
26. 2 5 50 and 97 5 quantiles Each of these quantities is joined to its direct neighbours as defined by axis by straight lines Solid red in the case of the median and dashed blue for the 95 posterior interval to form a piecewise linear curve the model fit In cases where other is specified its values are also plotted using black dots against the corresponding values of axis e g model fit eta 1 1 7 0 0 500 0 1 00E 3 1500 0 2 00E 3 Where appropriate either or both axes can be changed to a logarithmic scale via a property editor see Model fit plot scatterplot by default the posterior means of node are plotted using blue dots against the corresponding values of axis and an exponentially weighted smoother is fitted Numerous properties of a scatterplot can be modified using the editor described in Scatterplot for example the smoothing parameter may be changed or the smoother may be replaced by a different type of line or 95 posterior intervals for node may be displayed etc Correlations top home PA Correlation Tool beg fi end fi 000000 scatter matris print This non modal dialog box is used to plot out the relationship between the simulated values of selected variables which must have been monitored nodes scalars or arrays may be entered in each box and all combinations of variables entered in the two boxes are selected If a single array is given all pairwise correla
27. 4 No MC error is available on the DIC MC error on Dbar can be obtained by monitoring deviance and is generally quite small The primary concern is to ensure convergence of Dbar it is therefore worthwhile checking the stability of Doar over a long chain 5 The minimum DIC estimates the model that will make the best short term predictions in the same spirit as Akaike s criterion However if the difference in DIC is say less than 5 and the models make very different inferences then it could be misleading just to report the model with the lowest DIC 6 DICs are comparable only over models with exactly the same observed data but there is no need for them to be nested 7 DIC differs from Bayes factors and BIC in both form and aims 8 Caution is advisable in the use of DIC until more experience has been gained It is important to note that the calculation of DIC will be disallowed for certain models Please see the WinBUGS 1 4 web page for details http Awww mrc bsu cam ac uk bugs winbugs contents shtml Mixtures of models of different complexity top home Suppose we assume that each observation or group of observations is from one of a set of distributions where the members of the set have different complexity For example we may think data for each person s growth curve comes from either a linear or quadratic line We might think we would require reversible jump techniques but this is not the case as we are really only con
28. 7 8 9 10 intoa2 5 dimensional matrix in the order i jJth element of matrix value 1 1 1 1 2 2 1 3 3 1 5 5 2 1 6 2 5 10 whereas S Plus reads the same string of numbers in the order i jJth element of matrix value 1 1 1 2 1 2 1 2 3 1 3 5 2 3 6 2 5 10 Hence the ordering of the array dimensions must be reversed before using the S Plus deut command to create a data file for input into WinBUGS For example consider the 2 5 dimensional matrix ben ae en 1 2 3 4 5 Ki kl Kl Li Li The S Plus command gt dput list M M will then produce the following data file list NM structure Dim c 5 2 Data Edit the Dim statement in this file from format to input the required 2 5 dimensional matrix into WinBUGS file matrix dat GE 2 3 4 Dim c 5 2 to Now consider a 3 2 4 dimensional array 5 9 13 17 21 10 14 18 22 11 15 19 23 12 16 20 24 Bip Op Tp Spe 9p 10 This must be stored in S Plus as the 4 2 3 dimensional array gt A F F 1 1 2 1 1 5 DIR 2 6 EIER 3 7 4 4 8 r F 2 1 2 ars 9g 13 2 10 14 3 11 15 4 12 16 F F 3 1 2 1 17 21 2 18 22 3 19 23 4 20 24 The command gt dput list A A will then produce the following data file 14 15 16 17 18 list A structure Data 19
29. WinBUGS a scripting language has been provided This language can be useful for automating routine analysis The language works in effect by writing values into fields and clicking on buttons in relevant dialog boxes Note that it is possible to combine use of this scripting language with use of the menu dialog box interface To make use of the scripting language for a specific problem a minimum of four files are required the script itself a file containing the BUGS language representation of the model a file or several containing the data and for each chain a file containing initial values Each file may be in either native WinBUGS format odc or text format in which case it must have a txt extension The shortcut BackBUGS has been set up to run the commands contained in the file script odc in the root directory of WinBUGS when it is double clicked Thus a WinBUGS session may be embedded within any software component that can execute the BackBUGS shortcut Below is a list of currently implemented commands in the scripting language Alongside each is a brief synopsis of its menu dialog box equivalent first we have the menu name and then the associated menu option then an underlined name corresponds to a button in a dialog box whereas a name without an underline corresponds to a dialog box field set equal to the value of the quantity in italics that the gt points to If the menu dialog box equivalent of the specified script command wou
30. a Doodle Specification top home BZ Specification Tool check model load data compile num of chains fi load mits for chain fl gen inits This non modal dialog box acts on the focus view check model If the focus view contains text WinBUGS assumes the model is specified in the BUGS language The check model button parses the BUGS language description of the statistical model as in the classic version of BUGS If a syntax error is detected the cursor is placed where the error was found and a description of the error is given on the status line lower left corner of screen If a stretch of text is highlighted the parsing starts from the first character highlighted i e highlight the word mode1 else parsing starts at the top of the window If the focus view contains a Doodle i e the Doodle has been selected and is surrounded by a hairy border WinBUGS assumes the model has been specified graphically If a syntax error is detected the node where the error was found is highlighted and a description of the error is given on the status line load data The load data button acts on the focus view it will be greyed out unless the focus view contains text Data can be identified in two ways 1 if the data is in a separate document the window containing that document needs to be in the focus view the windows title bar will be coloured not grey when the load data command is used 2 if the data is spec
31. aie WinBUGS User Manual Version 1 4 January 2003 David Spiegelhalter Andrew Thomas Nicky Best Dave Lunn 1 MRC Biostatistics Unit Institute of Public Health Robinson Way Cambridge CB2 2SR UK 2 Department of Epidemiology amp Public Health Imperial College School of Medicine Norfolk Place London W2 1PG UK e mail bugs mrc bsu cam ac uk general andrew thomas ic ac uk technical internet http www mrc bsu cam ac uk bugs Permission and Disclaimer please click here to read the legal bit More informally potential users are reminded to be extremely careful if using this program for serious statistical analysis We have tested the program on quite a wide set of examples but be particularly careful with types of model that are currently not featured If there is a problem WinBUGS might just crash which is not very good but it might well carry on and produce answers that are wrong which is even worse Please let us know of any successes or failures Beware MCMC sampling can be dangerous Contents Introduction z This manual Advice for new users MCMC methods How WinBUGS syntax differs from that of ClassicBUGS Changes from WinBUGS 1 3 Compound Documents What is a compound document Working with compound documents Editing compound documents Compound documents and e mail Printing compound documents and Doodles Reading in text files Model Specification Graphical models Graphs as a fo
32. al CAR models moved to GeoBUGS new display options now possible to print out posterior correlation coefficients for monitored variables new manual sections Batch mode Scripts Tricks WinBUGS Graphics Tutorial and Changing MCMC Defaults 8 Compound Documents BUGS Contents What is a compound document Working with compound documents Editing compound documents Compound documents and e mail Printing compound documents and Doodles Reading in text files What is a compound document top home A compound document contains various types of information formatted text tables formulae plots graphs etc displayed in a single window and stored in a single file The tools needed to create and manipulate these information types are always available so there is no need to continuously move between different programs The WinBUGS software has been designed so that it produces output directly to a compound document and can get its input directly from a compound document To see an example of a compound document click here WinBUGS is written in Component Pascal using the BlackBox development framework see http www oberon ch In WinBUGS a document is a description of a statistical analysis the user interface to the software and the resulting output Compound documents are stored with the odc extension Working with compound documents top home A compound document is like a word processor document t
33. as p y theta where y comprises all stochastic nodes given values i e data and theta comprises the stochastic parents of y stochastic parents are the stochastic nodes upon which the distribution of y depends when collapsing over all logical relationships beg and end numerical fields are used to select a subset of the stored sample for analysis thin numerical field used to select every kth iteration of each chain to contribute to the statistics being calculated where k is the value of the field Note the difference between this and the thinning facility on the Update Tool dialog box when thinning via the Update Tool we are permanently discarding samples as the MCMC simulation runs whereas here we have already generated and stored a suitable number of posterior samples and may wish to discard some of them only temporarily Thus setting k gt 1 here will not have any impact on the storage memory requirements of processing long runs if you wish to reduce the number of samples actually stored to free up memory you should thin via the Update Tool chains to can be used to select the chains which contribute to the statistics being calculated clear removes the stored values of the variable from computer memory set must be used to start recording a chain of values for the variable trace plots the variable value against iteration number This trace is dynamic being redrawn each time the screen is redrawn history p
34. ate check box es show bars 95 per cent posterior intervals 2 5 97 5 for node will be shown on the plot as vertical bars if this box is checked show line use this check box to specify whether or not a reference line either a straight line specified via its intercept and gradient or an exponentially weighted smoother see below is to be displayed To the immediate right of the show line check box is a colour field that determines the reference line s colour if displayed linear or smooth these radio buttons are used to specify whether the reference line if it is to be displayed should be linear or whether an exponentially weighted smoother should be fitted instead In the case where a linear line is required its intercept and gradient should be entered in the two numeric fields to the right of the linear radio button in that order If a smoother is required instead then the desired degree of smoothing see below should be entered in the numeric field to the right of the smooth radio button To save unnecessary redrawing of the plot as the various numeric parameters are changed the Redraw line command button is used to inform WinBUGS of when all alterations to the parameters have been completed Note The exponentially weighted smoother that WinBUGS uses on scatterplots is defined as i ag aM i 1 H where n is the number of scattered points and the summations are from j 1 to j n The weights wj are given by
35. ated list of parents enclosed in brackets e g r dbin p n The distributions that can be used in WinBUGS are described in Distributions Clicking on the name of each distribution should provide a link to an example of its use provided with this release The parameters of a distribution must be explicit nodes in the graph scalar parameters can also be numerical constants and so may not be function expressions For distributions not featured in Distributions see Tricks Advanced Use of the BUGS Lanquage Censoring and truncation ep home Censoring is denoted using the notation I lower upper eg x ddist theta I lower upper would denote a quantity x from distribution ddist with parameters theta which had been observed to lie between Lower and upper Leaving either Lower or upper blank corresponds to no limit eg I lower corresponds to an observation known to lie above Lower Whenever censoring is specified the censored node contributes a term to the full conditional distribution of its parents This structure is only of use if x has not been observed if x is observed then the constraints will be ignored It is vital to note that this construct does NOT correspond to a truncated distribution which generates a likelihood that is a complex function of the basic parameters Truncated distributions might be handled by working out an algebraic form for the likelihood and using the techniques for arbitrary distrib
36. ates to be carried out refresh the number of updates between redrawing the screen thin the samples from every kth iteration will be stored where k is the value of thin Setting k gt 1 can help to reduce the autocorrelation in the sample but there is no real advantage in thinning except to reduce storage requirements and the cost of handling the simulations when very long runs are being carried out update Click to start updating the model Clicking on update during sampling will pause the simulation after the current block of iterations as defined by refresh has been completed the number of updates required can then be changed if needed Clicking on update again will restart the simulation This button becomes active when the model has been successfully compiled and given initial values iteration shows the total number of iterations stored after thinning not the actual number of iterations carried out In this respect updates represents the required number of samples rather than MCMC updates for example if 100 samples are requested via updates 100 and thin is set equal to 10 then 10 100 1000 iterations will actually be carried out of which 100 every 10th will be stored over relax click on this box a tick will then appear to select an over relaxed form of MCMC Neal 1998 which will be executed where possible This generates multiple samples at each iteration and then selects one that is negatively correlated with t
37. become highlighted in blue usually to indicate that the window is currently in focus Highlight the word mode 1 at the beginning of the code by dragging the mouse over the word whilst holding down the LMB Check the model syntax by clicking once with LMB on the check model button in the Specification Tool window A message saying model is syntactically correct should appear in the bottom left of the WinBUGS program window Step 3 We next need to load in the data The data can be represented using S Plus object notation file Manuals Tutorial seeds_S data odc or as a combination of an S Plus object and a rectangular array with labels at the head of each column file Manuals Tutorial seeds_mix_data odc Open one of the data files now To load the data in file Manuals Tutorial seeds S data odc Highlight the word 1ist at the beginning of the data file Click once with the LMB on the load data button in the Specification Tool window A message saying data loaded should appear in the bottom left of the WinBUGS program window To load the data in file Manuals Tutorial seeds_mix_data odc Highlight the word 1 ist at the beginning of the data file Click once with the LMB on the load data button in the Specification Tool window A message saying data loaded should appear in the bottom left of the WinBUGS program window Next highlight the whole of the header line i e column labels of the rectangular array data Click onc
38. could be badly distorted Remove Selection removes the highlighting from the selected node or plate of the Doodle if any Write Code opens a window containing the BUGS language equivalent to the Doodle No variable declaration statement is produced as it is not needed within WinBUGS After constructing a Doodle you are strongly recommended to use Write Code to check the structure is that which you intended Creating anode el home Point the mouse cursor to an empty region of the Doodle window and click A flashing caret appears next to the blue word name Typed characters will appear both at this caret and within the outline of the node Constant nodes can be given a name or a numerical value The name of a node starts with a letter and can also contain digits and the period character The name must not contain two successive periods and must not end with a period Vectors are denoted using a square bracket notation with indices separated by commas A colon separated pair of integers is used to denote an index range of a multivariate node or plate When first created the node will be of type stochastic and have associated density dnorm The type of the node can be changed by clicking on the blue word type at the top of the doodle A menu will drop down giving a choice of stochastic logical and constant for the node type Stochastic Associated with stochastic nodes is a density Click on the blue word density to see the choice of den
39. d then be used as an initial value file for future runs Seed top home Eefseed Tool xi set Opens a non modal dialog box containing seed a text entry field where the new seed of the random number generator can be typed coverage the pseudo random number generator used by WinBUGS generates a finite albeit very long sequence of distinct numbers which would eventually be repeated if the sampler were run for a sufficiently long time The coverage field shows the percentage of this sequence covered during the current WinBUGS session set sets the seed to the value entered into the dialog box The seed must be set after the model is checked via check model see Specification in order for the new value to apply to the current analysis Script top home The Script command is used to run model scripts in batch mode if the focus view contains a series of WinBUGS batch mode commands then selecting this command from the Model menu will cause the script to be executed Please see Batch mode Scripts for full details 8 The Inference Menu BUGS Contents General properties Samples Compare Correlations Summary Rank DIC General properties top home These menu items open dialog boxes for making inferences about parameters of the model The commands are divided into three sections the first three commands concern an entire set of monitored values for a variable the next two co
40. der to aid direct comparison of results Assuming the consequences of K prior distributions are to be compared a replicate the dataset K times within the model code b set up a loop to repeat the analysis for each prior holding results in arrays c compare results using the compare facility The example prior sensitivity explores six different suggestions for priors on the random effects variance in a meta analysis Modelling unknown denominators ep bone Suppose we have an unknown Binomial denominator for which we wish to express a prior distribution It can be given a Poisson prior but this makes it difficult to express a reasonably uniform distribution Alternatively a continuous distribution could be specified and then the round function used For example suppose we are told that a fair coin has come up heads 10 times how many times has it been tossed model r lt 10 p lt 0 5 r dbin p n n cont dunif 1 100 nc round n cont node mean sd MC error 2 5 median 97 5 start sample n 21 08 4 794 0 07906 13 0 21 0 32 0 1001 5000 n cont 21 08 4 804 0 07932 13 31 20 6 32 0 1001 5000 Assuming a uniform prior for the number of tosses we can be 95 sure that the coin has been tossed between 13 and 32 times A discrete prior on the integers could also have been used in this context Handling unbalanced datasets top nome Suppose we observe the following data on three individuals Person 1 13 2 Person 2 12 3
41. e E T K Se ic o 2710 3 4710 3 6 103 8710 S 104 iteration Trace of beetles 10000 values H BR A E g ZS 10 4 0 2 10 3 4 10 3 6103 810 3 iteration 25 50 75 Trace of beetles2 10000 values nena o 210 3 4 10 3 6 103 8 10 3 10 iteration Trace of beetles2 10000 values 0 210 S 4 10 3 6 10 S 8 10 3 10 4 iteration If you are running more than one chain simultaneously the trace and history plots will show each chain in a different colour In this case we can be reasonably confident that convergence has been achieved if all the chains appear to be overlapping one another The following plots are examples of i multiple chains for which convergence looks reasonable top and ii multiple chains which have clearly not reached convergence bottom alpha0 chains 1 2 iteration alphaO chains 1 2 iteration For a more formal approach to convergence diagnosis the software also provides an implementation see here of the techniques described in Brooks amp Gelman 1998 and a facility for outputting monitored samples in a format that is compatible with the CODA software see here How many iterations after convergence top home Once you are happy that convergence has been achieved you will need to run the simulation for a further number of iterations to obtain samples that can be used for posterior inference The more samples you save the more accurate will be your p
42. e same type rather than using the Al Plots tab we use a drag and pick facility First focus a single plot and select which font is to be modified for the whole group title axes or other if available Now highlight the group of plots using the mouse Hold down the ALT key and then the left hand mouse button and drag the mouse over an area of text with the desired font then release the mouse button and the ALT key in turn and the required changes should be made As an alternative to dragging the mouse over a piece of text with the desired font the user may instead drag over another plot even one in the group to be modified the group will adopt that plot s properties for the selected font on the Fonts tab Specific properties via Special top home Below we describe the special property editors that are available for certain types of plot these allow user interaction beyond that afforded by the Plot Properties dialogue box and are accessed via its Special button Density plot top home When the density button on the Sample Monitor Tool is pressed the output depends on whether the specified variable is discrete or continuous if the variable is discrete then a histogram is produced whereas if it is continuous a kernel density estimate is produced instead The specialized property editor that appears when Special on the Plot Properties dialogue box is selected also differs slightly depending on the nature of the s
43. e with the LMB on the load data button in the Specification Tool window A message saying data loaded should appear in the bottom left of the WinBUGS program window Step 4 Now we need to select the number of chains i e sets of samples to simulate The default is 1 but we will use 2 chains for this tutorial since running multiple chains is one way to check the convergence of your MCMC simulations Type the number 2 in the white box labelled num of chains in the Specification Tool window In practice if you have a fairly complex model you may wish to do a pilot run using a single chain to check that the model compiles and runs and obtain an estimate of the time taken per iteration Once you are happy with the model re run it using multiple chains say 2 5 chains to obtain a final set of posterior estimates Step 5 Next compile the model by clicking once with the LMB on the compile button in the Specification Tool window A message saying model compiled should appear in the bottom left of the WinBUGS program window This sets up the internal data structures and chooses the specific MCMC updating algorithms to be used by WinBUGS for your particular model Step 6 Finally the MCMC sampler must be given some initial values for each stochastic node These can be arbitrary values although in practice convergence can be poor if wildly inappropriate values are chosen You will need a different set of initial values for each chain i e two
44. ear predictor in probit regression too large indicates numerical overflow See possible solutions below for Trap undefined real result c logical expression too complex a logical node is defined in terms of too many parameters constants or too many operators try introducing further logical nodes to represent parts of the overall calculation for example a1 a2 a3 b1 b2 b3 could be written as A B where A and B are the simpler logical expressions a1 a2 a3 and b1 b2 b3 respectively Note that linear predictors with many terms should be formulated by vectorizing parameters and covariates and by then using the inprod function see Logical nodes d invalid or unexpected token scanned check that the value field of a logical node in a Doodle has been completed e unable to choose update method indicates that a restriction in the program has been violated see Restrictions when modelling above f undefined variable variables in a data file must be defined in a model just put them in as constants or with vague priors If a logical node is reported undefined the problem may be with a node on the right hand side g index out of range usually indicates that a loop index goes beyond the size of a vector or matrix dimension sometimes however appears if the has been omitted from the beginning of a comment line Some Trap messages top home a undefined real result
45. entially a random effects logistic regression allowing for over dispersion If p is the probability of germination on the ith plate we assume ri Binomial p nj logit p Og 4X4 AXo O14 9X4 Xo b b Normal 0 7 where x4 and Xa are the seed type and root extract of the ith plate and an interaction term Ou is included Specifying a model in the BUGS language top home The BUGS language allows a concise expression of the model using the twiddles symbol to denote stochastic probabilistic relationships and the left arrow lt sign followed by sign to denote deterministic logical relationships The stochastic parameters on 04 2 amp 42 and t are given proper but minimally informative prior distributions while the logical expression for sigma allows the standard deviation of the random effects distribution to be estimated model for 1 in 1 N r i dbin p i n i b i dnorm 0 tau logit p i lt alphaO alphal xl i alpha2 x2 i alphal2 xl i x2 i b i lpha0 dnorm 0 1 0E 6 lphal dnorm 0 1 0E 6 lpha2 dnorm 0 1 0E 6 alphal2 dnorm 0 1 0E 6 tau dgamma 0 001 0 001 sigma lt 1 sqrt tau DD o More detailed descriptions of the BUGS language along with lists of the available logical functions and stochastic distributions can be found in Model Specification and Distributions See also the on line examples
46. es gt node gt node clear stats node Inference gt Samples gt node gt node stats density node Inference gt Samples gt node gt node density autoC node Inference gt Samples gt node gt node auto cor trace node Inference gt Samples gt node gt node trace history node Inference gt Samples gt node gt node history quantiles node Inference gt Samples gt node gt node quantiles gr node Inference gt Samples gt node gt node bar diag coda node file stem Inference gt Samples gt node gt node coda set summary node stats summary node mean summary node clear summary node set rank node stats rank node hist rank node clear rank node Inference gt Summary Inference gt Summary Inference gt Summary Inference gt Summary Inference gt Rank gt node gt Inference gt Rank gt node gt Inference gt Rank gt node gt Inference gt Rank gt node gt gt node gt node set gt node gt node stats gt node gt node mean gt node gt node clear node set node stats node histogram node clear dic set Inference gt DIC gt set dic stats Inference gt DIC gt DIC quit File gt Exit save file 2 File gt Save As gt File name gt file Save script file 3 Model gt Script
47. g of data eg matrices in data files need to have the full structure format all data in datafile need to be described in the model need data list of constants and file sizes need column headings on rectangular arrays The data can be copied into the odc file or kept as a separate file c Copy the contents of the in file into the odc file Changes from WinBUGS 1 3 top home modular on line manual ability to run in batch mode using scripts running of default script on start up to allow calling from other programs new graphics see here for example and editing of graphics note that graphics from previous versions of the software will be incompatible with this version 1 4 missing data and range constraints allowed for multivariate normal new distributions negative binomial generalized gamma multivariate Student t DIC menu option for model comparison Options menu for advanced control of MCMC algorithms for example new syntax for more efficient inverse function interp lin interpolation function cut function recursively and thus efficiently calculated running quantiles MCMC algorithms block updating of fixed effects see here and or here for details non integer binomial and Poisson data Poisson as prior for continuous quantity coverage of random number generator additional restrictions END command for rectangular arrays spati
48. hat contains special rectangular embedded regions or elements each of which can be manipulated by standard word processing tools each rectangle behaves like a single large character and can be focused selected moved copied deleted etc If an element is focused the tools to manipulate its interior become available The WinBUGS software works with many different types of elements the most interesting of which are Doodles which allow statistical models to be described in terms of graphs DoodleBUGS is a specialised graphics editor and is described fully in DoodleBUGS The Doodle Editor Other elements are rather simpler and are used to display plots of an analysis Editing compound documents rop bone WinBUGS contains a built in word processor which can be used to manipulate any output produced by the software If a more powerful editing tool is needed WinBUGS documents or parts of them can be pasted into a standard OLE enabled word processor Text is selected by holding down the left mouse button while dragging the mouse over a region of text Warning if text is selected and a key pressed the selection will be replaced by the character typed The selection can be removed by pressing the Esc key or clicking the mouse A single element can be selected by clicking once into it with the left mouse button A selected element is distinguished by a thin bounding rectangle If this bounding rectangle contains small solid squares at the corne
49. he current problem All other components of the Updater options dialog box i e fields and command buttons pertain to the currently selected item in methods used for this text field simply describes what type of node the currently selected method is generally used for Note it is now possible to change the sampling methods for certain classes of distribution although this is delicate and should be done carefully please see Changing MCMC Defaults advanced users only for details iterations some updating algorithms entail iterative procedures that terminate when some relevant criterion is satisfied It is always possible however that within a given Gibbs iteration this criterion cannot be satisfied in reasonable time In such cases rather than allow the computer to hang it is preferable to specify a maximum number of iterations allowed before an error message is generated This maximum number of iterations is displayed in the iterations field which may be edited In cases where iterative procedures are not required the iterations field will be greyed out adaptive phase some updating methods such as Metropolis Hastings have an adaptive phase during which their internal parameters are tuned based on information gained from the chain s generated so far All samples generated during an adaptive phase should be discarded when drawing inferences but sometimes the default adaptive phase is longer than necessary meaning that the sa
50. he current value The time per iteration will be increased but the within chain correlations should be reduced and hence fewer iterations may be necessary However this method is not always effective and should be used with caution The auto correlation function may be used to check whether the mixing of the chain is improved adapting This box will be ticked while the Metropolis or slice sampling MCMC algorithm is in its initial tuning phase where some optimization parameters are tuned All summary statistics for the model will ignore information from this adapting phase The Metropolis and slice sampling algorithms have adaptive phases of 4000 and 500 iterations respectively which will be discarded from all summary statistics For details of how to change these default settings please see Update options Monitor Metropolis top home This command is only active if the Metropolis algorithm is being used for the model This shows the minimum maximum and average acceptance rate averaged over 100 iterations as the Metropolis algorithm adapts over the first N iterations The rate should lie between the two horizontal lines The first N iterations of the simulation cannot be used for statistical inference The default value of N is 4000 see Update options for details of how to change this value Save State top home Opens a window showing the current state of all the stochastic variables in the model displayed in S Plus format This coul
51. her interact with specific types of plot these are discussed below Margins top home The Margins tab displays the plot s left right top and bottom margins in millimetres mm The left and bottom margins are used for drawing the y and x axes respectively The top margin provides room for the plot s title and the right margin is typically used for plots that require a legend Note that top margins and hence titles are always inside the plotting rectangle i e there is no gap between the plotting rectangle and the top edge of the graphic BA Piot Properties 3 Margins Axis Bounds Titles All Plots Fonts left E right fo top E bottom fi2 In cases where it is not inappropriate to alter a plot s margins the user may enter his her preferred values and click on Apply to effect the desired change If the specified values are not appropriate e g if left right is greater than the width of the graphic which would result in a plotting rectangle of negative width or if any margin is negative etc then either nothing will happen and the Margins tab will reset itself or some form of compromise will be made Axis Bounds top home The user may specify new minimum and maximum values for either or both axes using the Axis Bounds tab followed by the Apply button BS Plot Properties xi Margins Axis Bounds Titles All Plots Fonts x min fi x max 1 0E 4 y min Go y max fio Apply Special N
52. hooting tips and troubleshooting advice for frequently experienced problems Tutorial a tutorial for new users Changing MCMC Defaults advanced users only how to change some of the default settings for the MCMC algorithms used in WinBUGS Distributions lists the various closed form distributions available in WinBUGS References references to relevant publications Users are advised that this manual only concerns the syntax and functionality of WinBUGS and does not deal with issues of Bayesian reasoning prior distributions statistical modelling monitoring convergence and so on If you are new to MCMC you are strongly advised to use this software in conjunction with a course in which the strengths and weaknesses of this procedure are described Please note the disclaimer at the beginning of this manual There is a large literature on Bayesian analysis and MCMC methods For further reading see for example Carlin and Louis 1996 Gelman et al 1995 Gilks Richardson and Spiegelhalter 1996 Brooks 1998 provides an excellent introduction to MCMC Chapter 9 of the Classic BUGS manual Topics in Modelling discusses non informative priors model criticism ranking measurement error conditional likelihoods parameterisation spatial models and so on while the CODA documentation considers convergence diagnostics Congdon 2001 shows how to analyse a very wide range of models using WinBUGS The BUGS website provides additional
53. ial observation How WinBUGS syntax differs from that of Classic BUGS top home Changes to the BUGS syntax have been kept as far as possible to simplifications There is now No need for constants these are declared as part of the data No need for variable declaration but all names used to declare data must appear in the model No need to specify files for data and initial values No limitation on dimensionality of arrays No limitation on size of problems except those dictated by hardware No need for semi colons at end of statements these were never necessary anyway A major change from the Classic BUGS syntax is that when defining multivariate nodes the range of the variable must be explicitly defined for example x 1 K dmnorm mu tau must be used instead of x dmnorm mu tau and for precision matrices you must write say tau 1 K 1 K dwish R 3 rather thantau dwish R 3 The following format must now be used to invert a matrix Sigma 1 K 1 K lt inverse taul Note that inverse is now a vector valued function as opposed to the relatively inefficient component wise evaluation required in previous versions of the software To convert Classic BUGS files to run under WinBUGS a Open the bug file as a text file delete unnecessary declarations and save as an odc document b Open dat files data has to be formatted as described in Formattin
54. ichardson S and Spiegelhalter D J Eds 1996 Markov chain Monte Carlo in Practice Chapman and Hall London UK Neal R 1997 Markov chain Monte Carlo methods based on slicing the density function Technical Report 9722 Department of Statistics University of Toronto Canada http www cs utoronto ca radford publications html Neal R 1998 Suppressing random walks in Markov chain Monte Carlo using ordered over relaxation In Learning in Graphical Models M Jordan ed Kluwer Academic Publishers Dordrecht pp 205 230 http www cs utoronto ca radford publications html Roberts G O 1996 Markov chain concepts related to sampling algorithms In W R Gilks S Richardson and D J Spiegelhalter Eds Markov chain Monte Carlo in Practice Chapman and Hall London UK Spiegelhalter D J Best N G Carlin B P and van der Linde A 2002 Bayesian measures of model complexity and fit with discussion J Roy Statist Soc B 64 583 640 Tierney L 1983 A space efficient recursive procedure for estimating a quantile of an unknown distribution SIAM J Sci Stat Comput 4 706 711
55. if requested blocks of fixed effect parameters in GLMs see above There is also the option of using ordered over relaxation Neal 1998 which generates multiple samples at each iteration and then selects one that is negatively correlated with the current value The time per iteration will be increased but the within chain correlations should be reduced and hence fewer iterations may be necessary However this method is not always effective and should be used with caution A slice sampling algorithm is used for non log concave densities on a restricted range This has an adaptive phase of 500 iterations which will be discarded from all summary statistics The current Metropolis MCMC algorithm is based on a symmetric normal proposal distribution whose standard deviation is tuned over the first 4000 iterations in order to get an acceptance rate of between 20 and 40 All summary statistics for the model will ignore information from this adapting phase It is possible for the user to change some aspects of the various available MCMC updating algorithms such as the length of an adaptive phase please see Update options for details It is also now possible to change the sampling methods for certain classes of distribution although this is delicate and should be done carefully see Changing MCMC Defaults advanced users only for details The shifted Poisson distribution occurs when a Poisson prior is placed on the order of a single binom
56. ified as part of a document the first character of the data either list if in S Plus format or the first array name if in rectangular format must be highlighted and the data will be read from there on See here for details of data formats Any syntax errors or data inconsistencies are displayed in the status line Corrections can be made to the data without returning to the check model stage When the data have been loaded successfully Data Loaded should appear in the status line The load data button becomes active once a model has been successfully checked and ceases to be active once the model has been successfully compiled num of chains The number of chains to be simulated can be entered into the text entry field next to the caption num of chains This field can be typed in after the model has been checked and before the model has been compiled By default one chain is simulated compile The compile button builds the data structures needed to carry out Gibbs sampling The model is checked for completeness and consistency with the data A node called deviance is automatically created which calculates minus twice the log likelihood at each iteration up to a constant This node can be monitored by typing deviance in the Samples dialog box This command becomes active once the model has been successfully checked and when the model has been successfully compiled model compiled should appear in the status line load inits
57. iles Ratsx and Ratsy See here for details of how to handle unbalanced data Note that programs exist for conversion of data from other packages please see the BUGS resources web page at http www mrc bsu cam ac uk bugs weblinks webresource shtml 8 DoodleBUGS The Doodle Editor BUGS Contents General properties Creating a node Selecting a node Deleting a node Moving a node Creating a plate Selecting a plate Deleting a plate Moving a plate Resizing a plate Creating an edge Deleting an edge Moving a Doodle Resizing a Doodle Printing a Doodle General properties top home Doodles consist of three elements nodes plates and edges The graph is built up out of these elements using the mouse and keyboard controlled from the Doodle menu The menu options are described below New opens a new window for drawing Doodles A dialog box opens allowing a choice of size of the Doodle graphic and the size of the nodes of the graph Grid the snap grid is on the centre of each node and each corner of each plate is constrained to lie on the grid Scale Model shrinks the Doodle so that the size of each node and plate plus the separation between them is reduced by a constant factor The Doodle will move towards the top left corner of the window This command is useful if you run out of space while drawing the Doodle Note that the Doodle will still be constrained by the snap grid and if the snap is coarse then the Doodle
58. ill colour of the displayed boxes Caterpillar plot top home The caterpillar plot property editor is virtually identical with the box plot property editor except that there is no fill colour field on the former B35 Caterpillar plot proper a IV show baseline 5 2490864320 IV show labels vert bar sh show means egestas show medians I rank I log scale Model fit plot top home In cases where an axis is defined on a strictly positive range it may be given a logarithmic scale by checking the appropriate check box Model fit properties Scatterplot top home With a scatterplot focused select Special on the Plot Properties dialogue box to obtain the following property editor Ze show means I log x scale show medians E Iog scale C DI I show bars IV show line UI DI linear smooth 14 99890 Redraw line show means or show medians these radio buttons determine whether it is the posterior means or medians of node that are plotted scattered against axis to form the plot that is node and axis on the Comparison Tool the default is means YA Scatterplot properties Immediately beneath the show means or show medians radio buttons is a colour field for selecting the colour of the scattered points log x log y scale either or both axes may be given a logarithmic scale assuming they are defined on strictly positive ranges by checking the appropri
59. it is positive This trick is illustrated by an example new sampling in which a normal likelihood is constructed using the zeros trick and compared to the standard analysis C lt 10000 this just has to be large enough to ensure all phi i s gt 0 for i in 1 N zeros i lt 0 phi i lt log L i Cc zeros i dpois phi i This trick allows arbitrary sampling distributions to be used and is particularly suitable when say dealing with truncated distributions A new observation x pred can be predicted by specifying it as missing in the data file and assigning it a uniform prior e g x pred dflat improper uniform prior on new x However our example shows that this method can be very inefficient and give a very high MC error An alternative to using zeros is the ones trick where the data is a set of 1 s assumed to be the results of Bernoulli trials with probabilities p i By making each p i proportional to L i i e by specifying a scaling constant large enough to ensure all p i s are lt 1 the required likelihood term is provided C lt 10000 this just has to be large enough to ensure all p i s lt 1 for i in 1 N ones i lt 1 p i lt L i C ones i dbern p i Specifying a new prior distribution top home If for a parameter theta say we want to use a prior distribution that is not part of the standard set then we can use the zeros trick See above at the p
60. known quantities Empirical summary statistics can be formed from these samples and used to draw inferences about their true values The sampling methods are used in the following hierarchies in each case a method is only used if no previous method in the hierarchy is appropriate Continuous target distribution Method Conjugate Direct sampling using standard algorithms Log concave Derivative free adaptive rejection sampling Gilks 1992 Restricted range Slice sampling Neal 1997 Unrestricted range Current point Metropolis Discrete target distribution Method Finite upper bound Inversion Shifted Poisson Direct sampling using standard algorithm In cases where the graph contains a Generalized Linear Model GLM component it is possible to request see Blocking options that WinBUGS groups or blocks together the fixed effect parameters and updates them via the multivariate sampling technique described in Gamerman 1997 This is essentially a Metropolis Hastings algorithm where at each iteration the proposal distribution is formed by performing one iteration starting at the current point of Iterative Weighted Least Squares IWLS If WinBUGS is unable to classify the full conditional for a particular parameter p say according to the above hierarchy then an error message will be returned saying Unable to choose update method for p Simulations are carried out univariately except for explicitly defined multivariate nodes and
61. ld normally be greyed out because of inappropriate timing for example then the script command will not execute and an error message will be produced instead script command menu dialog box equivalent display option Options gt Output options gt option window or log check model file Model gt Specification gt check model data data file blockfe aption compile chains inits chain inits file gen inits update iterations refresh every over relax option thin updater thin Model gt Specification gt load data Options gt Blocking options gt fixed effects gt option Model gt Specification gt num of chains gt chains compile Model gt Specification gt for chain gt chain load inits Model gt Specification gt gen inits Model gt Update gt updates gt iterations update Model gt Update gt refresh gt every Model gt Update gt over relax gt option Model gt Update gt thin gt thin beg iter Inference gt Samples gt beg gt iter end iter Inference gt Samples gt end gt iter first ter Inference gt Samples gt chains gt iter last iter Inference gt Samples gt to gt iter thin samples thin Inference gt Samples gt thin gt thin set node Inference gt Samples gt node gt node set clear node Inference gt Sampl
62. links to sites of interest some of which provide extensive examples and tutorial material Note that WinBUGS simulates each node in turn this can make convergence very slow and the program very inefficient for models with strongly related parameters such as hidden Markov and other time series structures If you have the educational version of WinBUGS you can run any model on the example data sets provided except possibly some of the newer examples If you want to analyse your own data you will only be able to build models with less than 100 nodes including logical nodes However the key for removing this restriction can be obtained by registering via the BUGS website from which the current distribution policy can also be obtained Advice for new users top home Although WinBUGS can be used without further reference to any of the BUGS project experience with using Classic BUGS may be an advantage and certainly the documentation on BUGS Version 0 5 and 0 6 available from http www mrc bsu cam ac uk bugs contains examples and discussion on wider issues in modelling using MCMC methods If you are using WinBUGS for the first time the following stages might be reasonable 1 Step through the simple worked example in the tutorial 2 Try other examples provided with this release see Examples Volume 1 and Examples Volume 2 3 Edit the BUGS language to fit an example of your own If you are interested in using Doodles 4 Try editing
63. lots out a complete trace for the variable The next six buttons will be greyed out if the MCMC simulation is in an adaptive phase density plots a smoothed kernel density estimate for the variable if it is continuous or a histogram if it is discrete auto cor plots the autocorrelation function of the variable out to lag 50 The values underlying these can be listed to a window by double clicking on the plot followed by cirl left mouse click if multiple chains are being simulated the values for each chain will be given Stats produces summary statistics for the variable pooling over the chains selected The required percentiles can be selected using the percentile selection box The quantity reported in the MC error column gives an estimate of N1 2 the Monte Carlo standard error of the mean The batch means method outlined by Roberts 1996 p 50 is used to estimate coda dumps out an ascii representation of the monitored values suitable for use in the CODA S Plus diagnostic package An output file for each chain is produced corresponding to the out files of CODA showing the iteration number and value to four significant figures There is also a file containing a description of which lines of the out file correspond to which variable this corresponds to the CODA ind file These can be named accordingly and saved as text files for further use Care may be required to stop the Windows system adding a t xt extension whe
64. methods Learning about the parameters of a Dirichlet distribution top home Suppose as part of a model there are J probability arrays pj 1 K j 1 J where K is the dimension of each array and sum p j 1 K 1 for all j We give each of them a Dirichlet prior pol 1 K ddirch alpha and we would like to learn about alpha However the parameters alphal of a Dirichlet distribution cannot be stochastic nodes The trick is to note that if delta k dgamma alpha k 1 then the vector with elements delta k sum delta 1 K k 1 K is Dirichlet with parameters alphalk k 1 K So the following construction should allow learning about the parameters alphal for k in 1 K pij k lt delta J k sum delta j delta j k dgamma alpha k 1 A prior can be put directly on the alphafk s Use of the cut function top home Suppose we observe some data that we do not wish to contribute to the parameter estimation and yet we wish to consider as part of the model This might happen for example a when we wish to make predictions on some individuals on whom we have observed some partial data that we do not wish to use for parameter estimation b when we want to use data to learn about some parameters and not others c when we want evidence from one part of a model to form a prior distribution for a second part of the model but we do not want feedback from this seco
65. mial distribution must be specified and cannot be given prior distributions There is however a trick to avoid this constraint for the Dirichlet distribution see here Structured precision matrices for multivariate normals these can be used in certain circumstances If a Wishart prior is not used for the precision matrix of a multivariate normal node then the elements of the precision matrix are updated univariately without any check of positive definiteness This will result in a crash unless the precision matrix is parameterised appropriately This is the user s responsibility Non integer data for Poisson and binomial Previously only integer valued data were allowed with Poisson and binomial distributions this restriction has now been lifted More generally it is now possible to specify a Poisson prior for any continuous quantity Range constraints using the I notation cannot be used with multivariate nodes except for multivariate normal distributions in which case the arguments to the I function may be specified as pblanks or as vector valued bounds Logical nodes top home Logical nodes are represented by the node name followed by a left pointing arrow followed by a logical expression of its parent nodes e g mu i lt betaO betal zl i beta2 z2 i b i Logical expressions can be built using the following operators plus multiplication minus division and unitary minus The
66. mmands are space saving short cuts that monitor running statistics and the final command DIC concerns evaluation of the Deviance Information Criterion proposed by Spiegelhalter et al 2002 Users should ensure their simulation has converged before using Summary Rank or DIC Note that if the MCMC simulation has an adaptive phase it will not be possible to make inference using values sampled before the end of this phase Samples top home BZ Sample Monitor Tool E node DI chains E to fi percentiles i bom i 5 25 clear set trace histori density a a story T J T 30 stats coda quanities bar dion auta cor 95 This command opens a non modal dialog for analysing stored samples of variables produced by the MCMC simulation The fields are node The variable of interest must be typed in this text field If the variable of interest is an array slices of the array can be selected using the notation variable lowerO upper0O lowerl upperl The buttons at the bottom of the dialog act on this variable A star can be entered in the node text field as shorthand for all the stored samples WinBUGS generally automatically sets up a logical node to measure a quantity known as deviance this may be accessed in the same way as any other variable of interest by typing its name i e deviance in the node field of the Sample Monitor Tool The definition of deviance is 2 log likelihood likelinood is defined
67. mpler is somewhat wasteful Alternatively the default adaptive phase may not be sufficiently long to allow proper tuning When the adaptive phase field is not greyed out indicating that the currently selected method in methods requires tuning it displays the length of the adaptive phase in iterations Gibbs cycles this may be edited by the user over relaxation many updating methods are capable of generating over relaxed samples Here at each iteration a number of candidate samples is generated and one that is negatively correlated with the current value is selected the time per iteration will be increased but within chain correlations should be reduced The number of candidate samples including the current value is displayed in the over relaxation field Set The Set command button applies the values shown in iterations adaptive phase and over relaxation to the updating method currently selected in methods for the current model if a new model is loaded or if WinBUGS is shut down and re started then the software will revert to its default values Save The Save button also applies the values shown in iterations adaptive phase and over relaxation to the currently selected method but it also saves those values as defaults for that method the next time that that method is used the new values will be selected automatically SKS Batch mode Scripts The Scripting Language As an alternative to the menu dialog box interface of
68. n saving enclosing the required file name in quotes should prevent this quantiles plots out the running mean with running 95 confidence intervals against iteration number bgr diag calculates the Gelman Rubin convergence statistic as modified by Brooks and Gelman 1998 The width of the central 80 interval of the pooled runs is green the average width of the 80 intervals within the individual runs is blue and their ratio R pooled within is red for plotting purposes the pooled and within interval widths are normalised to have an overall maximum of one The statistics are calculated in bins of length 50 R would generally be expected to be greater than 1 if the starting values are suitably over dispersed Brooks and Gelman 1998 emphasise that one should be concerned both with convergence of R to 1 and with convergence of both the pooled and within interval widths to stability The values underlying these plots can be listed to a window by double clicking on the plot followed by ctrl left mouse click on the window See WinBUGS Graphics for details of how to customize these plots Compare top home Select Compare from the Inference menu to open the Comparison Tool dialog box This is designed to facilitate comparison of elements of a vector of nodes with respect to their posterior distributions YA Comparison Tool node beg fi end aT other box plot caterpillar axis model fit scatterplot
69. nd part The cut function forms a kind of valve in the graph prior information is allowed to flow downwards through the cut but likelihood information is prevented from flowing upwards For example the following code leaves the distribution for theta unchanged by the observation y model y lt 2 y dnorm theta cut 1 theta cut lt cut theta theta dnorm 0 1 8 WinBUGS Graphics l BUGS Contents General properties Margins Axis Bounds Titles All Plots Fonts Specific properties via Special Density plot Box plot Caterpillar plot Model fit plot Scatterplot General properties top home All WinBUGS graphics have a set of basic properties that can be modified using the Plot Properties dialog box This is opened as follows first focus the relevant plot by left clicking on it then select Object Properties from the Edit menu Alternatively right clicking on a focused plot will reveal a pop up menu from which Properties may equivalently be selected The Plot Properties dialogue box comprises a tab view and two command buttons namely Apply and Special The tab view contains five tabs that allow the user to make different types of modification these are discussed below The Apply command button applies the properties displayed in the currently selected tab to the focused plot And Special opens any special property editors that have been designed so that the user may furt
70. ons cover the following topics Introduction the software and how a new user can start using WinBUGS Differences with previous incarnations of BUGS and WinBUGS are described Compound Documents the use of the compound document interface that underlies the program showing how documents can be created edited and manipulated Model Specification the role of graphical models and the specification of the BUGS language DoodleBUGS The Doodle Editor the DoodleBUGS software which allows complex Bayesian models to be specified as Doodles using a graphical interface The Model Menu the Mode Menu permits models expressed as either Doodles or in the BUGS language to be parsed checked and compiled The Inference Menu the nference Menu controls the monitoring display and summary of the simulated variables tools include specialized graphics and space saving short cuts for simple summaries of large numbers of variables The Info Menu the nfo menu provides a log of the run and other information The Options Menu facility that allows the user some control over where the output is displayed and the various available MCMC algorithms Baich mode Scripts how to run WinBUGS in batch mode using scripts Tricks Advanced Use of the BUGS Language special tricks for dealing with non standard problems e g specification of arbitrary likelinood functions WinBUGS Graphics how to display and change the format of graphical output Tips and Troubles
71. osterior estimates One way to assess the accuracy of the posterior estimates is by calculating the Monte Carlo error for each parameter This is an estimate of the difference between the mean of the sampled values which we are using as our estimate of the posterior mean for each parameter and the true posterior mean As arule of thumb the simulation should be run until the Monte Carlo error for each parameter of interest is less than about 5 of the sample standard deviation The Monte Carlo error MC error and sample standard deviation SD are reported in the summary statistics table see section Obtaining summaries of the posterior distribution Obtaining summaries of the posterior distribution top home The posterior samples may be summarised either graphically e g by kernel density plots or numerically by calculating summary statistics such as the mean variance and quantiles of the sample To obtain summaries of the monitored samples to be used for posterior inference Select Samples from the Inference menu Type the name of the parameter in the white box marked node or select the name from the pull down list or type star to select all monitored parameters Type the iteration number that you want to start your summary from in the white box marked beg this allows the pre convergence burn in samples to be discarded Click once with the LMB on the button marked stats a table reporting various summa
72. ote that the resulting minima and maxima may not be exactly the same as the values specified because WinBUGS always tries to ensure that axes range between nice numbers and also have a sufficient number of tick marks Note also that if max lt min is specified then WinBUGS will ignore it and the Axis Bounds tab will reset itself but there is no guard against specifying a range that does not coincide with the data being plotted and so the contents of the plot may disappear Some types of plot such as trace plots do not allow the user to change their axis bounds because it would be inappropriate to do so Titles top home The Titles tab should be self explanatory BS Plot Properties x Margins Axis Bounds Titles an Plots Fonts title muft x axis iteration y axis Note however that because WinBUGS does not yet support vertical text a substantial amount of space may be required in the left margin in order to write the y axis title horizontally if sufficient space is not available then the y axis title may not appear at all clearly this can be rectified by increasing the left margin All Plots top home The idea behind the AU Plots tab is to allow the user to apply some or all of the properties of the focused plot to all plots of the same type as the focused plot in the same window as the focused plot EAP Properties a Margins Axis Bounds Titles Al Plots Fonts Apply all applies cur
73. pecified variable In both cases the editor comprises a numeric field and two command buttons apply and apply all but in the case of a histogram the numeric field corresponds to the bin size whereas for kernel density estimates it is the smoothing parameter see below xd Seel bin size H smooth E apply apply all apply apply all property editor for histogram property editor for kernel density estimate In either case the apply button sets the bin size or smoothing parameter of the focused plot to the value currently displayed in the numeric field The apply all button on the other hand applies the value currently displayed in the numeric field to all plots of the same type as the focused plot in the same window as the focused plot Note We define the smoothing parameter for kernel density estimates via the definition of band width Suppose our posterior sample comprises m realisations of variable z denote these by z i 1 m band width V12 m9 v m z mE z where the summations are from i 1 to i m The default setting for is 0 2 Box plot top home The box plot property editor opened by pressing Special on the Plot Properties dialogue box when a box plot is focused is described as follows FA Box plot properties xi IV show baseline 52490864320 IV show labels Ze vert box C horiz box I log scale show means C show medians I rank ee E
74. posterior summaries of the model parameters you first need to set monitors for each parameter of interest This tells WinBUGS to store the values sampled for those parameters otherwise WinBUGS automatically discards the simulated values There are two types of monitor in WinBUGS Samples monitors Setting a samples monitor tells WinBUGS to store every value it simulates for that parameter You will need to set a samples monitor if you want to view trace plots of the samples to check convergence see Checking convergence or if you want to obtain exact posterior quantiles for example the posterior 95 Bayesian credible interval for that parameter Note that you can also obtain approximate 2 5 50 and 97 5 quantiles of the posterior distribution for each parameter using summary monitors To set a samples monitor Select Samples from the Inference menu Type the name of the parameter to be monitored in the white box marked node Click once with the LMB on the button marked set Repeat for each parameter to be monitored Summary monitors Setting a summary monitor tells WinBUGS to store the running mean and standard deviation for the parameter plus approximate running quantiles 2 5 50 and 97 5 The values saved contain less information than saving each individual sample in the simulation but require much less storage This is an important consideration when running long simulations e g 1000 s of iterations and storing
75. r ends correspond therefore to the 2 5 and 97 5 quantiles Note that this representation differs somewhat from the traditional The default value of the baseline shown on the plot is the global mean of the posterior means There is a special property editor available for box plots as indeed there is for all graphics generated via the Comparison Tool This can be used to interact with the plot and change the way in which it is displayed for example it is possible to rank the distributions by their means or medians and or plot them on a logarithmic scale For full details please see Box plot caterpillar a caterpillar plot is conceptually very similar to a box plot The only significant differences are that the inter quartile ranges are not shown and the default scale axis is now the x axis each distribution is summarised by a horizontal line representing the 95 interval and a dot to show where the mean is Again the default baseline in red is the global mean of the posterior means Due to their greater simplicity caterpillar plots are typically preferred over box plots when the number of distributions to be compared is large See Caterpillar plot for details of how to change the properties of caterpillar plots model fit the elements of node and other if specified are treated as a time series defined by increasing values of the elements of axis The posterior distribution of each element of node is summarised by the
76. rent settings of selected items to all plots of same type in current document margins title E x bounds J7 x axis title y bounds y axis title Apply all The user should first configure the focused plot in the desired way and then decide which of the plot s various properties are to be applied to all plots of the same type in the same window The user should then check all relevant check boxes and click on Apply all Be careful It is easy to make mistakes here and there is no undo option the best advice for users who go wrong is to reproduce the relevant graphics window via the Sample Monitor Tool or the Comparison Tool for example Fonts top home The Fonts tab allows the user to change the font of the plot s title its axes and any other text that may be present BS Plot Properties x Margins Axis Bounds Titles All Plots Fonts Click on Modify to choose a new font forthe selected item Ze title C axes pies Modify First select the font to be modified and click on Modify Note that it will only be possible to select other if the focused plot has a third font i e if text other than the title and axes is present on the plot e g the labels on a box plot see Compare The self explanatory Font dialogue box should appear select the required font and click on OK or Cancel In order to apply the same font to an arbitrarily large group of plots not necessarily of th
77. rior level A single zero Poisson observation with mean phi phi theta contributes a term exp phi to the likelihood for theta when this is combined with a flat prior for theta the correct prior distribution results zero lt 0 theta dflat phi lt expression for log desired prior for theta zero dpois phi This is illustrated in new prior by an example in which a normal prior is constructed using the zeros trick and the results are compared to the standard formulation It is important to note that this method produces high auto correlation poor convergence and high MC error so it is computationally slow and long runs are necessary Specifying a discrete prior on a set of values top home Suppose you want a parameter D to take one of a set of values d 1 d K say with probabilities p 1 p K Then specify the arrays d 1 K and p 1 K in a data file or in the model and use M deat p choose which element of d to use D lt da M This is illustrated by an example t df of learning about the degrees of freedom of a t distribution If the discrete prior is put on too coarse a grid then there may be numerical problems crashes unless good initial values are provided and very poor mixing It is therefore advised to either use a continuous prior or a discrete prior on a fine grid see the t df example Using pD and DIC top home Here we make a number of observations regarding
78. rmal lanquage The BUGS language stochastic nodes Censoring and truncation Constraints on using certain distributions Logical nodes Arrays and indexing Repeated structures Data transformations Nested indexing and mixtures Formatting of data DoodleBUGS The Doodle Editor z General properties Creating a node Selecting a node Deleting a node Moving a node Creating a plate Selecting a plate Deleting a plate Moving a plate Resizing a plate Creating an edge Deleting an edge Moving a Doodle Resizing a Doodle Printing a Doodle The Model Menu z General properties Specification Update Monitor Metropolis Save State Seed Script The Inference Menu General properties Correlations Rank DIC The Info Menu z General properties Open Log Clear Log Node info Components The Options Menu Output options Blocking options Update options Batch mode Scripts Tricks Advanced Use of the BUGS Language Specifying a new sampling distribution Specifying a new prior distribution Specifying a discrete prior on a set of values Using pD and DIC Mixtures of models of different complexity Where the size of a set is a random quantity Assessing sensitivity to prior assumptions Modelling unknown denominators Handling unbalanced datasets Learning about the parameters of a Dirichlet distribution Use of the cut function WinBUGS Graphics General properties Margins A
79. rs and mid sides it can be resized by dragging these with the mouse An element can be focused by clicking twice into it with the left mouse button A focused element is distinguished by a hairy grey bounding rectangle A selection can be moved to a new position by dragging it with the mouse To copy the selection hold down the control key while releasing the mouse button These operations work across windows and across applications and so the problem specification and the output can both be pasted into a single document which can then be copied into another word processor or presentation package The style size font and colour of selected text can be changed using the Attributes menu The vertical offset of the selection can be changed using the Text menu The formatting of text can be altered by embedding special elements The most common format control is the ruler pick option Show Marks in menu Text to see what rulers look like The small black up pointing triangles are tab stops which can be moved by dragging them with the mouse and removed by dragging them outside the left or right borders of the ruler The icons above the scale control for example centering and page breaks Vertical lines within tables can be curtailed by inserting a ruler and removing the lines by selecting each tab stop and then ctrl left mouse click Warning removing the left most line requires care there is a tab stop hidden behind the upper left most one tha
80. rt y as dependent variables without creating a separate variable z sqrt y in the data file The BUGS language therefore permits the following type of structure to occur for i in 1 N z i lt sqrt y i z i dnorm mu tau Strictly speaking this goes against the declarative structure of the model specification with the accompanying exhortation to construct a directed graph and then to make sure that each node appears once and only once on the left hand side of a statement However a check has been built in so that when finding a logical node which also features as a stochastic node such as z above a stochastic node is created with the calculated values as fixed data We emphasise that this construction is only possible when transforming observed data not a function of data and parameters with no missing values This construction is particularly useful in Cox modelling and other circumstances where fairly complex functions of data need to be used It is preferable for clarity to place the transformation statements in a section at the beginning of the model specification so that the essential model description can be examined separately See the Leuk and Endo examples Nested indexing and mixtures top home Nested indexing can be very effective For example suppose N individuals can each be in one of groups and g 1 N is a vector which contains the group membership Then group coefficients betafi can be fitted
81. ry statistics based on the sampled values of the selected parameter will appear Click once with the LMB on the button marked density a window showing kernel density plots based on the sampled values of the selected parameter will appear Please see the manual entry Samples for a detailed description of the Sample Monitor Tool dialog box i e that which appears when Samples is selected from the Inference menu v Changing MCMC Defaults advanced users only d Contents Defaults for numbers of iterations Defaults for sampling methods This section shows how to change some of the defaults for the MCMC algorithms used in WinBUGS Users do so at their own risk and should make a back up copy of the relevant default file first although in case of disaster a new copy of WinBUGS can always be downloaded The program should be restarted after any edits Defaults for numbers of iterations top home Options for iteration numbers can be changed temporarily using the Options menu see here for details but the default values are held in the WinBUGS file System Rsrc Registry odc which may be edited Examples include UpdaterMetnormal_AdaptivePhase default 4000 is the length of the adaptive phase of the general normal proposal Metropolis algorithm UpdaterSlice_AdaptivePhase default 500 is the adaptive phase of the slice sampling algorithm UpdaterSlice_Iterations default 100000 is how many tries the slice sampling algori
82. s in which WinBUGS will not permit the calculation of DIC and so the menu option is greyed out Please see the WinBUGS 1 4 web page for current restrictions http Awww mrc bsu cam ac uk bugs winbugs contents shtm set starts calculating DIC and related statistics the user should ensure that convergence has been achieved before pressing set as all subsequent iterations will be used in the calculation clear if a DIC calculation has been started via set this will clear it from memory so that it may be restarted later DIC displays the calculated statistics as described below please see Spiegelhalter et a 2002 for full details the section Tricks Advanced Use of the BUGS Lanquage also contains some comments on the use of DIC The DIC button generates the following statistics Dbar this is the posterior mean of the deviance which is exactly the same as if the node deviance had been monitored see here This deviance is defined as 2 log likelinood likelinood is defined as p y theta where y comprises all stochastic nodes given values i e data and theta comprises the stochastic parents of y stochastic parents are the stochastic nodes upon which the distribution of y depends when collapsing over all logical relationships Dhat this is a point estimate of the deviance 2 log likelinood obtained by substituting in the posterior means theta bar of theta thus Dhat 2 log p y theta bar
83. sformation facility described in Data transformations Constants These need to be specified in the data file If they only appear in value statements they do not need to be explicitly represented in the Doodle Selecting anode ep home Point the mouse cursor inside the node and left mouse click Deleting anode top home Select a node and press ctrl delete key combination Moving a node top home Select a node and then point the mouse into selected node Hold mouse button down and drag node The cursor keys may also be used Creating a plate top home Point the mouse cursor to an empty region of the Doodle window and click while holding the ctrl key down Selecting a plate top bone Point the mouse into the lower or right hand border of the plate and click Deleting a plate top home Select a plate and press ctrl delete key combination Moving a plate top home Select a plate and then point the mouse into the lower or right hand border of the selected plate Hold the mouse button down and drag the plate Resizing a plate top home Select a plate and then point the mouse into the small region at the lower right where the two borders intercept Hold the mouse button down and drag to resize the plate Creating an edge ron home Select node into which the edge should point and then click into its parent while holding down the ctrl key Deleting an edge top home Select node whose
84. sidering a single mixture model as a sampling distribution Thus standard methods for setting up mixture distributions can be adopted but with components having different numbers of parameters The mixtures example illustrates how this is handled in WinBUGS using a set of simulated data Naturally the standard warnings about mixture distributions apply in that convergence may be poor and careful parameterisation may be necessary to avoid some of the components becoming empty Where the size of a set is a random quantity top home Suppose the size of a set is a random quantity this naturally occurs in changepoint problems where observations up to an unknown changepoint K come from one model and after K come from another Note that we cannot use the construction for i in 1 K yli model 1 for i in K 1 N f yli model 2 since the index for a loop cannot be a random quantity Instead we can use the step function to set up an indicator as to which set each observation belongs to for i in 1 N ind i lt 1 step i K 0 01 will be 1 for all i lt K 2 otherwise yli modelind i This is illustrated in random sets by the problem of adding up terms in a series of unknown length and in Stagnant by a changepoint problem Assessing sensitivity to prior assumptions top home One way to do this is to repeat the analysis under different prior assumptions but within the same simulation in or
85. sities available this will not necessarily include all those available in the BUGS language and described in Distributions For each density the appropriate name s of the parameters are displayed in blue For some densities default values of the parameters will be displayed in black next to the parameter name When edges are created pointing into a stochastic node these edges are associated with the parameters in a left to right order To change the association of edges with parameters click on one of the blue parameter names a menu will drop down from which the required edge can be selected This drop down menu will also give the option of editing the parameter s default value Logical Associated with logical nodes is a link which can be selected by clicking on the blue word ink Logical nodes also require a value to be evaluated modified by the chosen link each time the value of the node is required To type input in the value field click the mouse on the blue word value The value field of a logical node corresponds to the right hand side of a logical relation in the BUGS language and can contain the same functions The value field must be completed for all logical nodes We emphasise that the value determines the value of the node and the logical links in the Doodle are for cosmetic purposes only It is possible to define two nodes in the same Doodle with the same name one as logical and one as stochastic in order to use the data tran
86. t can cause a crash if deleted in the usual way it seems to require a ctr right mouse click Compound documents and e mail top home WinBUGS compound documents contain non ascii characters but the Too s menu contains a command Encode Document which produces an ascii representation of the focus document The original document can be recovered from this encoded form by using the Decode command of the Tools menu This allows for example Doodles to be sent by e mail Printing compound documents and Doodles top home These can be printed directly from the File menu If postscript versions of Doodles or whole documents are wanted you could install a driver for a postscript printer say Apple LaserWriter but set it up to print to file checking the paper size is appropriate Alternatively Doodles or documents could be copied to a presentation or word processing package and printed from there Reading in text files top home Open these from the File menu as text files They can be copied into documents or stored as documents 8 Model Specification S BUGS Contents Graphical models Graphs as a formal lanquage The BUGS language stochastic nodes Censoring and truncation Constraints on using certain distributions Logical nodes Arrays and indexing Repeated structures Data transformations Nested indexing and mixtures Formatting of data Graphical models top home We strongly recommend that the first s
87. tep in any analysis should be the construction of a directed graphical model Briefly this represents all quantities as nodes in a directed graph in which arrows run into nodes from their direct influences parents The model represents the assumption that given its parent nodes pa vj each node vis independent of all other nodes in the graph except descendants of v where descendant has the obvious definition Nodes in the graph are of three types 1 Constants are fixed by the design of the study they are always founder nodes i e do not have parents and are denoted as rectangles in the graph They must be specified in a data file 2 Stochastic nodes are variables that are given a distribution and are denoted as ellipses in the graph they may be parents or children or both Stochastic nodes may be observed in which case they are data or may be unobserved and hence be parameters which may be unknown quantities underlying a model observations on an individual case that are unobserved say due to censoring or simply missing data 3 Deterministic nodes are logical functions of other nodes Quantities are specified to be data by giving them values in a data file in which values for constants are also given The BUGS language stochastic nodes top home In the text based model description stochastic nodes are represented by the node name followed by a twiddles symbol followed by the distribution name followed by a comma separ
88. thm has before it gives up and produces an error message Defaults for sampling methods top home It is now possible to change the sampling methods for certain classes of distribution although this is delicate and should be done carefully The sampling methods are held in Updater Rsrc Methods odc and can be edited For example if there are problems with WinBUGS adaptive rejection sampler DFreeARS then the method UpdaterDFreeARS for log concave could be replaced by UpdaterSlice normally used for real non linear this has been known to sort out some Traps However take care and don t forget to keep a copy of the original Methods odc file 8 Distributions lt lt BUGS Contents Discrete Univariate Bernoulli Binomial Categorical Negative Binomial Poisson Continuous Univariate Beta Chi squared Double Exponential Exponential Gamma Generalized Gamma Log normal Logisitic Normal Pareto Student t Uniform Weibull Discrete Multivariate D Multinomial Continuous Multivariate Dirichlet Multivariate Normal Multivariate Student t Wishart Discrete Univariate Bernoulli r dbern p Binomial r dbin p n Categorical E e d at pill Negative Binomial x dnegbin p r Poisson r dpois lambda Continuous Univariate Beta p dbeta a b Chi squared cx r 1 top home p p r 0 1 n r n r jaan EE rE pir r 1 2
89. tions will be plotted scatter produces a scatter plot of the individual simulated values matrix produces a matrix summary of the cross correlations print opens a new window containing the coefficients for all possible correlations among the selected variables The calculations may take some time Summary top home Summary MonitorTool node DI set stats mean clear This non modal dialog box is used to calculate running means standard deviations and quantiles The commands in this dialog are less powerful and general than those in the Sample Monitor Tool but they also require much less storage an important consideration when many variables and or long runs are of interest node The variable of interest must be typed in this text field set starts recording the running totals for node Stats displays the running means standard deviations and 2 5 50 median and 97 5 quantiles for node Note that these running quantiles are calculated via an approximate algorithm see here for details and should therefore be used with caution means displays the running means for node in a comma delimited form This can be useful for passing the results to other statistical or display packages clear removes the running totals for node Rank top home B83 Rank Monitor Tool xi node m percentiles 5 set stats Ys histogram clear D This non modal dialog box is used to store
90. trace plots of the sample values versus iteration to look for evidence of when the simulation appears to have stabilised To obtain live trace plots for a parameter Select Samples from the Inference menu Type the name of the parameter in the white box marked node Click once with the LMB on the button marked trace an empty graphics window will appear on screen Repeat for each parameter of interest Once you start running the simulations using the Update tool from the Model menu trace plots for these parameters will appear live in the graphics windows To obtain a trace plot showing the full history of the samples for any parameter for which you have previously set a samples monitor and carried out some updates Select Samples from the Inference menu Type the name of the parameter in the white box marked node or select the name from the pull down list of currently monitored nodes click once with the LMB on the downward facing arrowhead to the immediate right of the node field Click once with the LMB on the button marked history a graphics window showing the sample trace will appear Repeat for each parameter of interest The following plots are examples of i chains for which convergence in the pragmatic sense looks reasonable left hand side and ii chains which have clearly not reached convergence right hand side BEETLES Trace of beetles1 10000 values S ba Ka E
91. tted to appear directly as an index term intermediate deterministic nodes may be introduced if such functions are required The conventions broadly follow those of S Plus nim represents n n 1 M x represents all values of a vector x y 3 indicates all values of the third column of a two dimensional array y Multidimensional arrays are handled as one dimensional arrays with a constructed index Thus functions defined on arrays must be over equally spaced nodes within an array forexample sum i 1 4 k When dealing with unbalanced or hierarchical data a number of different approaches are possible see Handling unbalanced datasets The ideas discussed in Nested indexing and mixtures may also be helpful in this respect the user should bear in mind however the contiguous elements restriction described in Constraints on using certain distributions Repeated structures top home Repeated structures are specified using a for loop The syntax for this is for i in a b 4 list of statements to be repeated for increasing values of loop variable i Note that neither a nor b may be stochastic see here for a possible way to get round this Data transformations top home Although transformations of data can always be carried out before using WinBUGS it is convenient to be able to try various transformations of dependent variables within a model description For example we may wish to try both y and sq
92. un Tutorial BUGS Contents Introduction Specifying a model in the BUGS language Running a model in WinBUGS Monitoring parameter values Checking convergence How many iterations after convergence Obtaining summaries of the posterior distribution Introduction top home This tutorial is designed to provide new users with a step by step guide to running an analysis in WinBUGS It is not intended to be prescriptive but rather to introduce you to the main tools needed to run an MCMC simulation in WinBUGS and give some guidance on appropriate usage of the software The Seeds example from Volume of the WinBUGS examples will be used throughout this tutorial This example is taken from Table 3 of Crowder 1978 and concerns the proportion of seeds that germinated on each of 21 plates arranged according to a 2 by 2 factorial layout by seed and type of root extract The data are shown below where ri and nj are the number of germinated and the total number of seeds on the ith plate i 1 N These data are also analysed by for example Breslow and Clayton 1993 seed O aegyptiaco 75 seed O aegyptiaco 73 Bean Cucumber Bean Cucumber r n r n r n r n r n r n r n r n 10 39 0 26 5 6 0 83 8 16 0 50 3 12 0 25 23 62 0 37 53 74 0 72 10 30 0 33 22 41 0 54 23 81 0 28 55 72 0 76 8 28 0 29 15 30 0 50 26 51 0 51 32 51 0 63 23 45 0 51 32 51 0 63 17 39 0 44 46 79 0 58 0 4 0 00 3 7 0 43 10 13 0 77 The model is ess
93. used to represent the node this is for debugging purposes only it is of no practical use to the user Components ep home Displays all the components dynamic link libraries in use 8 The Options Menu A BUGS Contents Output options Blocking options Update options Output options top home YA Output options window log k H output precision window or log if window is selected then a new window will be opened for each new piece of output statistics traces etc if log is selected then all output will be pasted into a single log file output precision the number of significant digits in numerical output Blocking options top home Y A Blocking options 3 x If the box is checked then where possible WinBUGS will use the multivariate updating method described in Gamerman 1997 to generate new values for blocks of fixed effect parameters Update options top home Once a model has been compiled the various updating algorithms required in order to perform the MCMC simulation may be tuned somewhat via the Updater options dialog box select Update options from the Options menu PA updater options xj methods UpdaterGamma iterations UpdaterNormal adaptive phase 4000 over telaxation used for reai non linear Set Save methods this selection box shows the system names of all updating methods required selected for t
94. utions described in Tricks Advanced Use of the BUGS Language It is also important to note that if x theta lower and upper are all unobserved then Lower and upper must not be functions of theta Constraints on using certain distributions top home Contiguous elements Multivariate nodes must form contiguous elements in an array Since the final element in an array changes fastest such nodes must be defined as the final part of any array For example to define a set of K K Wishart variables as a single multidimensional array x i j k we could write for 1 in 1 1 Xli 1 K 1 K dwish R i 3 where R i is an array of specified prior parameters No missing data Data defined as multinomial or as multivariate Student t must be complete in that missing values are not allowed in the data array We realise this is an unfortunate restriction and we hope to relax it in the future For multinomial data it may be possible to get round this problem by re expressing the multivariate likelinood as a sequence of conditional univariate binomial distributions Note that multivariate normal data may now be specified with missing values Conjugate updating Dirichlet and Wishart distributions can only be used as parents of multinomial and multivariate normal nodes respectively Parameters you can t learn about and must specify as constants The parameters of Dirichlet and Wishart distributions and the order N of the multino
95. w to do this for the seeds example you should find that at least 2000 iterations are required for convergence Once you re happy that your simulation has converged you will need to run some further updates to obtain a sample from the posterior distribution The section How many iterations after convergence provides tips on deciding how many more updates you should run For the seeds example try running a further 10000 updates Once you have run enough updates to obtain an appropriate sample from the posterior distribution you may summarise these samples numerically and graphically see section Obtaining summaries of the posterior distribution for details on how to do this For the seeds example summary statistics for the monitored parameters are shown below node mean sd MC error 2 5 median 97 5 start sample alphaO 0 5553 0 1904 0 003374 0 9365 0 5563 0 1763 2001 20000 alpha 0 08693 0 3123 0 006873 0 5502 0 09157 0 6964 2001 20000 alpha12 0 8358 0 4388 0 01105 1 731 0 8253 0 002591 2001 20000 alpha2 1 359 0 2744 0 005812 0 8204 1 354 1 923 2001 20000 sigma 0 2855 0 146 0 005461 0 04489 0 2752 0 6132 2001 20000 To save any files created during your WinBUGS run focus the window containing the information you want to save and select the Save As option from the File menu To quit WinBUGS select the Exit option from the File menu Monitoring parameter values ep home In order to check convergence and obtain
96. x symmetric amp positive definite ei tek E References Best N G Cowles M K and Vines S K 1997 CODA Convergence diagnosis and output analysis software for Gibbs sampling output Version 0 4 MRC Biostatistics Unit Cambridge http www mrc bsu cam ac uk bugs classic coda04 readme shtml Breslow N E and Clayton D G 1993 Approximate inference in generalized linear mixed models Journal of the American Statistical Association 88 9 25 Brooks S P 1998 Markov chain Monte Carlo method and its application The Statistician 47 69 100 Brooks S P and Gelman A 1998 Alternative methods for monitoring convergence of iterative simulations Journal of Computational and Graphical Statistics 7 434 455 Carlin B P and Louis T A 1996 Bayes and Empirical Bayes Methods for Data Analysis Chapman and Hall London UK Congdon P 2001 Bayesian Statistical Modelling John Wiley amp Sons Chichester UK Crowder M J 1978 Beta binomial Anova for proportions Applied Statistics 27 34 37 Gamerman D 1997 Sampling from the posterior distribution in generalized linear mixed models Statistics and Computing 7 57 68 Gelman A Carlin J C Stern H and Rubin D B 1995 Bayesian Data Analysis Chapman and Hall New York Gilks W 1992 Derivative free adaptive rejection sampling for Gibbs sampling In Bayesian Statistics 4 J M Bernardo J O Berger A P Dawid and A F M Smith eds Oxford University Press UK pp 641 665 Gilks W R R
97. xis Bounds Titles All Plots Fonts Specific properties via Special Density plot Box plot Caterpillar plot Model fit plot Scatterplot Tips and Troubleshooting z Restrictions when modelling Some error messages Some Trap messages The program hangs Speeding up sampling Improving convergence Tutorial z Introduction Specifying a model in the BUGS language Running a model in WinBUGS Monitoring parameter values Checking convergence How many iterations after convergence Obtaining summaries of the posterior distribution Changing MCMC Defaults advanced users only Defaults for numbers of iterations Defaults for sampling methods Distributions Discrete Univariate Continuous Univariate Discrete Multivariate Continuous Multivariate References o e Introduction a Y BUGS Contents This manual Advice for new users MCMC methods How WinBUGS syntax differs from that of Classic BUGS Changes from WinBUGS 1 3 This manual top home This manual describes the WinBUGS software an interactive Windows version of the BUGS program for Bayesian analysis of complex statistical models using Markov chain Monte Carlo MCMC techniques WinBUGS allows models to be described using a slightly amended version of the BUGS language or as Doodles graphical representations of models which can if desired be translated to a text based description The BUGS language is more flexible than the Doodles The secti

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