JAGS Version 3.1.0 user manual
Contents
1. Beware however that every node in the data statement will be considered as data in the sub sequent model statement This example although superficially similar has a quite different interpretation data for i in 1 N y i dnorm mu tau mu dnorm 0 1 tau dgamma 1 3 model 4 for i in 1 N y i dnorm mu tau 33 mu dnorm 0 1 0E 3 tau dgamma 1 0E 3 1 0E 3 Since the names mu and tau are used in both data and model blocks these nodes will be considered as observed in the model and their values will be fixed at those values generated in the data block 7 0 5 Directed cycles Directed cycles are forbidden in JAGS There are two important instances where directed cycles are used in BUGS e Defining autoregressive priors e Defining ordered priors For the first case the GeoBUGS extension to OpenBUGS provides some convenient ways of defining autoregressive priors These should be available in a future version of JAGS 7 0 6 Censoring truncation and prior ordering These are three closely related issues that are all handled using the I construct in BUGS Censoring occurs when a variable X is not observed directly but is observed only to lie in the range L U Censoring is an a posteriori restriction of the data and is represented in OpenBUGS by the I construct e g X dnorm theta tau I L U where L and U are constant nodes Truncation occurs when a variable is known a pri2. If X dfoo theta T L U then a priori X is known to lie between L and U This generates a likelihood p z 9 P L lt X lt U 6 if L lt X lt U and zero otherwise where p x 0 is the density of X given 6 according to the distribution foo Note that calculation of the denominator may be computationally expensive For compatibility with OpenBUGS JAGS permits the use of 1 for truncation when the the parameters of the truncated distribution are fixed For example this is permitted mu dnorm 0 1 0E 3 ICO because the truncated distribution has fixed parameters mean 0 precision 1 0E 3 In this case there is no difference between censoring and truncation Conversely this is not permit ted for i in 1 N 4 x i dnorm mu tau I 0 mu dnorm 0 1 0E 3 tau dgamma 1 1 because the mean and precision of x1 zy are parameters to be estimated JAGS does not know if the aim is to model truncation or censoring and so the compiler will reject the model Use either T or the dinterval distribution to resolve the ambiguity Prior ordering of top level parameters in the model can be achieved using the sort func tion which sorts a vector in ascending order Symmetric truncation relations like this alpha 11 dnorm 0 1 0E 3 I alpha 2 alpha 2 dnorm 0 1 0E 3 I alpha 11 alpha 3 alpha 3 dnorm 0 1 0E 3 I alpha 21 Should be replaced by this for i in 1 3 alphaO i dnorm 0 1
3. methods that are efficient but limited to certain specific model structures and ending with the most generic possibly inefficient methods If no suitable Sampler can be generated for one of the model parameters an error message is generated The user has no direct control over the process of choosing Samplers However you may indirectly control the process by loading a module that defines a new Sampler Factory The module will insert the new Sampler Factory at the beginning of the list where it will be queried before all of the other Sampler Factories You can also optionally turn on and off sampler factories using the SET FACTORY command See 3 1 18 A report on the samplers chosen by the model and the stochastic nodes they act on can be generated using the SAMPLERS TO command See section 3 1 13 2 4 Adaptation and burn in In theory output from an MCMC sampler converges to the target distribution i e the posterior distribution of the model parameters in the limit as the number of iterations tends to infinity In practice all MCMC runs are finite By convention the MCMC output is divided into two parts an initial burn in period which is discarded and the remainder of the run in which the output is considered to have converged sufficiently close to the target distribution Samples from the second part are used to create approximate summary statistics for the target distribution By default JAGS keeps only the
4. aed eon Ko Kade ae So ile MODULES ties Skater Meee Ao aaa a Sate Bee Saldo LISE FACTORIES 250 ESA hee Gate ie A Ae eae Th fate Glare Sable SET FACTORY Li a Gata ety NA A A ae eat 3 14 19 MODEL CLEAR blo oo aS oe a Do ee Bode a ey SCE YH WAAAN OTK KR KL wo EH 3 1 20 Print Working Directory PWD 3 1 21 Change Directory CD bs ave tice pole a woke Heck dz Directory st DER ig A te eae te a ah a LD RUN ai ts Sate E Dado GE 2 ETEOLS a a oe yest a cance ae SE AS cat Sane te he Ss A ae ANE Modules 4 1 The base module ATA Base Samplers cesc uk wee e Ged pe Mk we AP MCD Base RN GS moir arana be ae atin rey Rak ob ney ea le Bee ee at ee a te at Alao Base Monitors di yt DG A artes te Ra Wee el BG 4 2 The bugs module ippo ea ke ek a a ee ea Be ae as 4 3 The mix Module erisin Slew a eg aE RS SS 4 4 The dic module 0 0 0 0 00 ee a 4 4 1 The deviance monitor 44 2 The pD monitor s sws p s ieo Dataa ea os P Oe ap oa a a a a as 4 4 3 The popt monitor a saaa aa e eee 4 5 Themsmmodule 0 0 2 ee 4 6 The glm module 0 0 2 eee ee ee ee Functions DL Basefunctions sa aa ae bee a Ree ad bee ER owe ae d 5 2 Functions in the bugs module 0000000004 521 Scalar functions gt oo La we Rages Bo a o 5 2 2 Scalar valued functions with vector arguments 5 2 3 Vector and array valued functions 2 4 Bid Function Aliases eco cee a is Se Sob wee ols be O Distribut
5. holder for any function with a name starting and ending with the character Such functions are automatically recognized as infix operators by the JAGS model parser with precedence defined by table 5 1 22 Type Usage Description Logical x Il y Or operators x amp amp y And Ix Not Comparison x gt y Greater than operators x gt y Greater than or equal to x lt y Less than x lt y Less than or equal to x y Equal Arithmetic x y Addition operators x y Subtraction x y Multiplication x y Division x fspecial y User defined operators l tad Unary minus Power function xy Table 5 1 Base functions listed in reverse order of precedence 5 2 Functions in the bugs module 5 2 1 Scalar functions Table 5 2 lists the scalar valued functions in the bugs module that also have scalar arguments These functions are automatically vectorized when they are given vector matrix or array arguments with conforming dimensions Table 5 4 lists the link functions in the bugs module These are smooth scalar valued func tions that may be specified using an S style replacement function notation So for example the log link log y lt x is equivalent to the more direct use of its inverse the exponential function y lt exp x This usage comes from the use of link functions in generalized linear models Table 5 3 shows functions to calculate the probability density probability function and quantiles of
6. some of the distributions provided by the bugs module These functions are parameterized in the same way as the corresponding distribution For example if x has a normal distribution with mean u and precision T x dnorm mu tau Then the usage of the corresponding density probability and quantile functions is density x lt dnorm x mu tau Density of normal distribution at x prob x lt pnorm x mu tau P X lt x quantile90 x lt qnorm 0 9 mu tau 90th percentile For details of the parameterization of the other distributions see tables 6 1 and 6 2 23 Usage Description Value Restrictions on arguments abs x Absolute value Real arccos x Arc cosine Real l lt a lt l arccosh x Hyperbolic arc cosine Real l lt zx arcsin x Arc sine Real l lt a lt l arcsinh x Hyperbolic arc sine Real arctan x Arc tangent Real arctanh x Hyperbolic arc tangent Real l lt a lt l cos x Cosine Real cosh x Hyperbolic Cosine Real cloglog x Complementary log log Real Usas 1 equals x y Test for equality Logical exp x Exponential Real icloglog x Inverse complementary Real log log function ilogit x Inverse logit Real log x Log function Real 20 logfact x Log factorial Real z gt l loggam x Log gamma Real a0 logit x Logit Real 0 lt a lt l phi x Standard normal cdf Real pow x z Power function Real If x lt 0 then z is integer probit x Probit Real 0 lt a lt l round x Round to integer Integer away fro
7. the right hand side These are referred to as the parent nodes Taken together the nodes in the model together with the parent child relationships represented as directed edges form a directed acyclic graph The very top level nodes in the graph with no parents are constant nodes which are defined either in the model definition e g 1 0E 3 or in the data file e g x 1 Relations can be of two types A stochastic relation defines a stochastic node repre senting a random variable in the model A deterministic relation lt defines a deterministic node the value of which is determined exactly by the values of its parents Nodes defined by a relation are embedded in named arrays Array names may contain letters numbers decimal points and underscores but they must start with a letter The node array mu is a vector of length N containing N nodes mu 1 mu N The node array alpha is a scalar JAGS follows the S language convention that scalars are considered as vectors of length 1 Hence the array alpha contains a single node alpha 1 Deterministic nodes do not need to be embedded in node arrays The node Y i could equivalently be defined as Y i dnorm alpha beta x i x bar tau In this version of the model definition the node previously defined as mu il still exists but is not accessible to the user as it does not have a name This ability to hide determinis tic nodes by embedding them in other expressions u
8. you get a logic error Error messages may also be generated when parsing files model files data files command files The error messages generated in this case are created automatically by the program bison They generally take the form syntax error unexpected FOO expecting BAR and are not always abundantly clear If a model compiles and initializes correctly but an error occurs during updating then the current state of the model will be dumped to a file named jags dumpN R where N is the chain number You should then load the dumped data into R to inspect the state of each chain when the error occurred 17 Chapter 4 Modules The JAGS library is distributed along with certain dynamically loadable modules that extend its functionality A module can define new objects of the following classes 1 functions and distributions the basic building blocks of the BUGS language 2 samplers the objects which update the parameters of the model at each iteration and sampler factories the objects that create new samplers for specific model structures If the module defines a new distribution then it will typically also define a new sampler for that distribution 3 monitors the objects that record sampled values for later analysis and monitor factories that create them 4 random number generators the objects that drive the MCMC algorithm and RNG factories that create them The base module and the bugs module are loaded automa
9. 0E 3 alpha 1 3 lt sort alpha0 35 Chapter 8 Feedback Please send feedback to martyn_plummer users sourceforge net I am particularly inter ested in the following problems e Crashes including both segmentation faults and uncaught exceptions e Incomprehensible error messages e Models that should compile but don t e Output that cannot be validated against OpenBUGS e Documentation erors If you want to send a bug report it must be reproducible Send the model file the data file the initial value file and a script file that will reproduce the problem Describe what you think should happen and what did happen 36 Chapter 9 Acknowledgments Many thanks to the BUGS development team without whom JAGS would not exist Thanks also to Simon Frost for pioneering JAGS on Windows and Bill Northcott for getting JAGS on Mac OS X to work Kostas Oikonomou found many bugs while getting JAGS to work on Solaris using Sun development tools and libraries Bettina Gruen Chris Jackson Greg Ridgeway and Geoff Evans also provided useful feedback Special thanks to Jean Baptiste Denis who has been very diligent in providing feedback on JAGS 37 Bibliography 1 James Albert and Siddharta Chib Bayesian analysis of binary and polychotomous re sponse data Journal of the American Statistical Association 88 669 679 1993 Gilles Celeux Merrilee Hurn and Christian P Robert Computational and inferential difficult
10. Chapter 7 Differences between JAGS and OpenBUGS Although JAGS aims for the same functionality as OpenBUGS there are a number of important differences 7 0 1 Data format There is no need to transpose matrices and arrays when transferring data between R and JAGS since JAGS stores the values of an array in column major order like R and FORTRAN i e filling the left hand index first If you have an S style data file for OpenBUGS and you wish to convert it for JAGS then use the command bugs2jags which is supplied with the coda package 7 0 2 Distributions Structural zeros are allowed in the Dirichlet distribution If p ddirch alpha and some of the elements of alpha are zero then the corresponding elements of p will be fixed to zero The Multinomial dmulti and Categorical dcat distributions which take a vector of probabilities as a parameter may use unnormalized probabilities The probability vector is normalized internally so that Pi y Pj Pi gt 7 0 3 Observable Functions Logical nodes in the BUGS language are a convenient way of describing the relationships between observables constant and stochastic nodes but are not themselves observable You cannot supply data values for a logical node This restriction can occasionally be inconvenient as there are important cases where the data are a deterministic function of unobserved variables Two important examples are 31 1 Censored data which
11. JAGS Version 3 1 0 user manual Martyn Plummer 6 August 2011 Contents 1 Introduction 2 Running a model in JAGS 21 Definitii 22a eee ee wk RN ee Re eb Wet A eS 2 1 1 Model definition 0 020200 0002 eee eee Did De Data ole a e a a Shc Ada dh octet gk Ma es 2 1 3 Node Array dimensions a hve Gee a 2 25 COMPA e hcl taeda ee aad ee Gee qed a E eae aAa 23 Anitializationy sec ae e icas Pie e MED AS a Ri Fos Ae eas Pee ek ve 2 31 Parameter Valles tit A e a A ee ie a N 23 CENSO A Pe ee Ata se a 2 353 Samplers ccna Cle oy PR Aa RE ae eee ee 8 aS 2 4 Adaptation and burda 2 9 Monitoring 654 55 ae a A oe Bed So Stok bossa a 3 Running JAGS 3 1 Scripting commands 1 2 daa ee ee ALO MODELIN a e hn he a e de cha eee ee a IE DATAIN a O it rh EM See Goel ote eg LS GOMPILE etree Sete ean A ape Rack el ate ee ot Pantie ee ae E Deke PARAMETERS UN 2g a of oe a Rey weer Sede ences Go he a a oro UNSEAT Pr E PARR ge BOW OR OBE SR lee ERA Se UPDATE e aa ngap Sidon SOR BNE AE de ea Ada gh ee a SLE ADAPT e doy pe es eds dag os obese Bund Be Solio MONITOR abans ape DL met st ee ho lee a le aot ATS Bel CODA tigate homie aca Seed wan Bled Ge whee A aa Sle DU AA A ate BOE en OO ae tal E Se DATA TO So a eS Pt amp oleae Dig he OS ROSS 3412 PARAMETERS TO a Me ee lg i Male Ae a ela 3 1 13 SAMPLERS TO tic e a e Mi RDP SR E E AA fhe Serio atk tera eee Me EEE Uhre Aes le hs Bae boy Sao MU NONI ca din oh eee
12. R Read data in from file compile nchains 2 Compile a model with two parallel chains parameters in line inits R Read initial values from file initialize Initialize the model update 1000 Adaptation if necessary and burnin for 1000 iterations monitor alpha Set trace monitor for node alpha monitor beta and beta monitor sigma and sigma update 10000 Update model for 10000 iterations coda All monitored values are written out to file More examples can be found in the file classic bugs tar gz which is available from Source forge http sourceforge net projects mcmc jags files Output from JAGS is printed to the standard output even when a script file is being used 3 1 Scripting commands JAGS has a simple set of scripting commands with a syntax loosely based on Stata Commands are shown below preceded by a dot This is the JAGS prompt Do not type the dot in when you are entering the commands C style block comments taking the form can be embedded anywhere in the script file Additionally you may use R style single line comments starting with If a scripting command takes a file name then the name may be optionally enclosed in quotes Quotes are required when the file name contains space or any character which is not alphanumeric or one of the following _ 12 In the descriptions below angular brackets lt gt and the text inside them represents a parameter th
13. S command 3 1 19 MODEL CLEAR model clear Clears the current model The data table see section 3 1 2 remains intact 3 1 20 Print Working Directory PWD pwd Prints the name of the current working directory This is where JAGS will look for files when the file name is given without a full path e g mymodel bug 3 1 21 Change Directory CD cd lt dirname gt Changes the working directory to lt dirname gt 16 3 1 22 Directory list DIR dir Lists the files in the current working directory 3 1 23 RUN run lt cmdfile gt Opens the file lt cmdfile gt and reads further scripting commands until the end of the file Note that if the file contains an EXIT statement then the JAGS session will terminate 3 2 Errors There are two kinds of errors in JAGS runtime errors which are due to mistakes in the model specification and logic errors which are internal errors in the JAGS program Logic errors are generally created in the lower level parts of the JAGS library where it is not possible to give an informative error message The upper layers of the JAGS program are supposed to catch such errors before they occur and return a useful error message that will help you diagnose the problem Inevitably some errors slip through Hence if you get a logic error there is probably an error in your input to JAGS although it may not be obvious what it is Please send a bug report see Feedback below whenever
14. al valued distributions in the bugs module 29 Name Usage Density Lower Upper Beta dbetabin a b n BAN A A E 0 n binomial a gt 0 b gt 0 n N z nx n Bernoulli dbern p io o 0 1 dep lt l pel p Binomial dbin p n DS gin Ema 0 n 0 lt p lt lneN 70 p Categorical dcat pi To 1 N TE REY Dai Ti Noncentral dhyper n1 n2 m1 psi 71 A max 0 n m1 min n m1 hypergeometric 0 lt ni 0 lt M lt n4 D EA Gna Negative dnegbin p r E Ac ooo 0 binomial 0 lt p lt i reNt z p Cae Poisson dpois lambda exp A A 0 A gt 0 q Table 6 2 Discrete univariate distributions in the bugs module Name Usage Density Dirichlet p ddirch alpha pj fa aj gt 0 IO 04 IL Tay Multivariate x dmnorm mu Omega io gt 7 normal Q positive definite 77 erpi e u Q x u 2 Wishart Omega dwish R k Q 6 D 2 RIK exp Tr ROQ 2 R p x p pos def k gt p 2Pk 2T k 2 Multivariate x dmt mu Omega k T k p 2 011 2 1 T te Rese Cee ce O Q x 2 Student t Q pos def TEID nr pegis PES A 1 Multinomial x dmulti pi n Ti n Sa sE Table 6 3 Multivariate distributions in the bugs module Distribution Canonical Alias Compatibile name with Binomial dbin dbinom R Chi square dchisqr dchisq R Negative binomial dnegbin dnbinom R Weibull dweib dweibull R Dirichlet ddirch ddirich OpenBUGS Table 6 4 Distributions with aliases in bugs module 30
15. at should be replaced with the correct value by you Anything inside square brackets is optional Do not type the square brackets if you wish to use an option 3 1 1 MODEL IN model in lt file gt Checks the syntactic correctness of the model description in file and reads it into memory The next compilation statement will compile this model See also MODEL CLEAR 3 1 19 3 1 2 DATA IN data in lt file gt JAGS keeps an internal data table containing the values of observed nodes inside each node array The DATA IN statement reads data from a file into this data table Several data statements may be used to read in data from more than one file If two data files contain data for the same node array the second set of values will overwrite the first and a warning will be printed See also DATA TO 3 1 11 3 1 3 COMPILE compile nchains lt n gt Compiles the model using the information provided in the preceding model and data state ments By default a single Markov chain is created for the model but if the nchains option is given then n chains are created Following the compilation of the model further DATA IN statements are legal but have no effect A new model statement on the other hand will replace the current model 3 1 4 PARAMETERS IN parameters in lt file gt chain lt n gt Reads the values in file and writes them to the corresponding parameters in chain n The file has the same format as the
16. commonly occurs in survival analysis In the most general case we know that unobserved failure time T lies in the interval L U 2 Aggregate data when we observe the sum of two or more unobserved variables JAGS contains two novel distributions to handle these situations 1 The dinterval distribution represents interval censored data It has two parameters t the original continuous variable and c a vector of cut points of length M say If X dinterval t c then X 0 if t lt cl X m if cm lt t lt clm 1 forl lt m lt M X M if cM lt t 2 The dsum distribution represents the sum of two or more variables It takes a variable number of parameters If Y dsum x1 x2 x3 then Y z1 z2 23 These distributions exist to give a likelihood to data that is in fact a deterministic function of the parameters The relation Y dsum x1 x2 is logically equivalent to Y lt xi x2 But the latter form does not create a contribution to the likelihood and does not allow you to define Y as data The likelihood function is trivial it is 1 if the parameters are consistent with the data and 0 otherwise The dsum distribution also requires a special sampler which can currently only handle the case where the parameters of dsum are unobserved stochastic nodes and where the parameters are either all discrete valued or all continuous valued A node cannot be subject to more than one dsum constraint 7 0 4 Data transformations JAGS all
17. current value of each node in the model unless a monitor has been defined for that node The burn in period of a JAGS run is therefore the interval between model initialization and the creation of the first monitor When a model is initialized it may be in adaptive mode meaning that the Samplers used by the model may modify their behaviour for increased efficiency Since this adaptation may depend on the entire sample history the sequence generated by an adapting sampler is no longer a Markov chain and is not guaranteed to converge to the target distribution Therefore adaptive mode must be turned off at some point during burn in and a sufficient number of iterations must take place after the adaptive phase to ensure successful burnin By default adaptive mode is turned off half way through first update of a JAGS model All samplers have a built in test to determine whether they have converged to their optimal sampling behaviour If any sampler fails this validation test a warning will be printed To ensure optimal sampling behaviour the model should be run again from scratch using a longer adaptation period The adapt command see section 3 1 7 can be used for more control over the adaptive phase The adapt command updates the model but keeps it in adaptive mode At the end of each update the convergence test is called The message Adaptation successful will be printed if the convergence test is successful otherwise the message will rea
18. d Adapation incomplete Successive calls to adapt are possible while keeping the model in adaptive mode The next update command will immediately turn off adaptive mode 2 5 Monitoring A monitor in JAGS is an object that records sampled values The simplest monitor is a trace monitor which stores the sampled value of a node at each iteration JAGS cannot monitor a node unless it has been defined in the model file For vector or array valued nodes this means that every element must be defined Here is an example of a simple for loop that only defines elements 2 to N of theta for i in 2 N 4 theta i dnorm 0 1 Unless theta 1 is defined somewhere else in the model file the multivariate node theta is undefined and therefore it will not be possible to monitor theta as a whole In such cases you can request each element separately e g theta 2 theta 3 etc or request a subset that is fully defined e g theta 2 6 Monitors can be classified according to whether they pool values over iterations and whether they pool values over parallel chains The standard trace monitor does neither When monitor values are written out to file using the CODA command the output files cre ated depend on the pooling of the monitor as shown in table 2 1 By default all of these files have the prefix CODA but this may be changed to any other name using the stem option to the CODA command See 3 1 9 The standard CODA format for monitor
19. d measures 4 4 1 The deviance monitor The deviance monitor records the deviance of the model i e the sum of the deviances of all the observed stochastic nodes at each iteration The command monitor deviance will create a deviance monitor unless you have defined a node called deviance in your model In this case you will get a trace monitor for your deviance node 4 4 2 The pD monitor The pD monitor is used to estimate the effective number of parameters pp of the model 11 It requires at least two parallel chains in the model but calculates a single estimate of pp across all chains 9 A pD monitor can be created using the command monitor pD Like the deviance monitor however if you have defined a node called pD in your model then this will take precedence and you will get a trace monitor for your pD node Since the pp monitor pools its value across all chains its values will be written out to the index file CODAindex0 txt and output file CODAoutput0 txt when you use the CODA command The effective number of parameters is the sum of separate contributions from all observed stochastic nodes pp pp There is also a monitor that stores the sample mean of pp These statistics may be used as influence diagnostics 11 The mean monitor for pp is created with monitor pD type mean Its values can be written out to a file PDtable0 txt with coda pD type mean stem PD 4 4 3 The popt monito
20. er product Dimensions of a b conform interp lin e vi v2 Linear Interpolation e scalar v1 v2 conforming vectors logdet a Log determinant a is a square matrix max x1 x2 Maximum element among all arguments mean x Mean of elements of a min x1 x2 Minimum element among all arguments prod x Product of elements of a sum a Sum of elements of a sd a Standard deviation of elements of a Table 5 5 Scalar valued functions with general arguments in the bugs module Usage Description Restrictions inverse a Matrix inverse a is a symmetric positive definite matrix mexp a Matrix exponential a is a square matrix rank v Ranks of elements of v vis a vector sort v Elements of v in order vis a vector t a Transpose a is a matrix a h h b Matrix multiplication a b conforming vector or matrices Table 5 6 Vector or matrix valued functions in the bugs module 5 2 2 Scalar valued functions with vector arguments Table 5 5 lists the scalar valued functions in the bugs module that take general arguments Unless otherwise stated in table 5 5 the arguments to these functions may be scalar vector or higher dimensional arrays The max and min functions work like the corresponding R functions They take a variable number of arguments and return the maximum minimum element over all supplied arguments This usage is compatible with OpenBUGS although more general 5 2 3 Vector and array valued functions Table 5 6 lists
21. f you rely on automatic initial value generation and are running multiple parallel chains then the initial values will be the same in all chains You may not want this behaviour especially if you are using the Gelman and Rubin convergence diagnostic which assumes that the initial values are over dispersed with respect to the posterior distribution In this case you are advised to set the starting values manually using the parameters in statement 2 3 2 RNGs Each chain in JAGS has its own random number generator RNG RNGs are more correctly referred to as pseudo random number generators They generate a sequence of numbers that merely looks random but is in fact entirely determined by the initial state You may optionally set the name of the RNG and its initial state in the initial values file The name of the RNG is set as follows RNG name lt name There are four RNGs supplied by the base module in JAGS with the following names pase Wichmann Hill base Marsaglia Multicarry base Super Duper base Mersenne Twister There are two ways to set the starting state of the RNG The simplest is to supply an integer value to RNG seed e g RNG seed lt 314159 The second is way to save the state of the RNG from one JAGS session see the PARAM ETERS TO statement section 3 1 12 and use this as the initial state of a new chain The state of any RNG in JAGS can be saved and loaded as an integer vector w
22. he samplers to the given file The output appears in three tab separated columns with one row for each sampled node e The index number of the sampler starting with 1 The index number gives the order in which Samplers are updated at each iteration e The name of the sampler matching the index number e The name of the sampled node If a Sampler updates multiple nodes then it is represented by multiple rows with the same index number 15 3 1 14 LOAD load lt module gt Loads a module into JAGS see chapter 4 Loading a module does not affect any previously initialized models but will affect the future behaviour of the compiler and model initialization 3 1 15 UNLOAD unload lt module gt Unloads a module Currently initialized models are unaffected but the functions distribution and factory objects in the model will not be accessible to future models 3 1 16 LIST MODULES list modules Prints a list of the currently loaded modules 3 1 17 LIST FACTORIES list factories type lt factype gt List the currently loaded factory objects and whether or not they are active The type option must be given and the three possible values of lt factype gt are sampler monitor and rng 3 1 18 SET FACTORY set factory lt facname gt lt status gt type lt factype gt Sets the status of a factor object The possible values of lt status gt are on and off Possible factory names are given from the LIST MODULE
23. ies with mixture posterior distributions Journal of the American Statistical Association 95 957 970 1999 T A Davis and W W Hager Modifying a sparse cholesky factorization SIAM Journal on Matriz Analysis and Applications 20 606 627 1999 Timothy A Davis Direct Methods for Sparse Linear Systems SIAM 2006 Sylvia Frihwirth Schnatter Rudolf Fr hwirth Leonhard Held and H vard Rue Im proved auxiliary mixture sampling for hierarchical models of non gaussian data Statistics and Computing 19 4 479 492 2009 Dani Gamerman Efficient sampling from the posterior distribution in generalized linear mixed models Statistics and Computing 7 57 68 1997 Chris Holmes and Leonard Held Bayesian auxiliary variable models for binary and multinomial regression Bayesian Analysis 1 1 145 168 2006 Radford Neal Sampling from multimodal distributions using tempered transitions Statistics and Computing 6 353 366 1994 M Plummer Discussion of the paper by Spiegelhalter et al Journal of the Royal Statis tical Society Series B 64 620 2002 M Plummer Penalized loss functions for Bayesian model comparison Biostatistics 9 3 523 539 2008 DJ Spiegelhalter NG Best BP Carlin and A van der Linde Bayesian measures of model complexity and fit with discussion Journal of the Royal Statistical Societey Series B 64 583 639 2002 38
24. ined Any node that is used on the right hand side of a relation but is not defined on the left hand side of any relation is assumed to be a constant node Its value must be supplied in the data file 3 The model uses a function or distribution that has not been defined in any of the loaded modules The number of parallel chains to be run by JAGS is also defined at compilation time Each parallel chain should produce an independent sequence of samples from the posterior distri bution By default JAGS only runs a single chain 2 3 Initialization Before a model can be run it must be initialized There are three steps in the initialization of a model 1 The initial values of the model parameters are set 2 A Random Number Generator RNG is chosen for each parallel chain and its seed is set 3 The Samplers are chosen for each parameter in the model 2 3 1 Parameter values The user may supply an initial value file containing values for the model parameters The file may not contain values for logical or constant nodes The format is the same as the data file see section 2 1 2 If initial values are not supplied by the user then each parameter chooses its own initial value based on the values of its parents The initial value is chosen to be a typical value from the prior distribution The exact meaning of typical value depends on the distribution of the stochastic node but is usually the mean median or mode I
25. ions 6 1 Distribution aliases a Differences between JAGS and OpenBUGS 7 0 1 Dataformat 7 0 2 Distributions 7 0 3 Observable Functions 7 0 4 Data transformations 7 0 5 Directed cycles 7 0 6 Censoring truncation and prior ordering Feedback Acknowledgments 16 16 17 17 17 18 18 18 19 19 19 19 20 20 20 20 21 21 22 22 23 23 26 26 26 28 28 31 31 31 31 32 34 34 36 37 Chapter 1 Introduction JAGS is Just Another Gibbs Sampler It is a program for the analysis of Bayesian models using Markov Chain Monte Carlo MCMC which is not wholly unlike OpenBUGS http www openbugs info JAGS was written with three aims in mind to have an engine for the BUGS language that runs on Unix to be extensible allowing users to write their own functions distributions and samplers and to be a platform for experimentation with ideas in Bayesian modelling JAGS is designed to work closely with the R language and environment for statistical computation and graphics http www r project org You will find it useful to install the coda package for R to analyze the output You can also use the rjags package to work directly with JAGS from within R but note that the rjags package is not described in this manual JAGS is licensed under the GNU General Public License version 2 You may freely modify and redistribute it under certain conditi
26. ith the name RNG state For example RNG state lt as integer c 20899 10892 29018 is a valid state for the Marsaglia Multicarry generator You cannot supply an arbitrary integer to RNG state Both the length of the vector and the permitted values of its elements are determined by the class of the RNG The only safe way to use RNG state is to re use a previously saved state If no RNG names are supplied then RNGs will be chosen automatically so that each chain has its own independent random number stream The exact behaviour depends on which modules are loaded The base module uses the four generators listed above for the first four chains then recycles them with different seeds for the next four chains and so on By default JAGS bases the initial state on the time stamp This means that when a model is re run it generates an independent set of samples If you want your model run to be reproducible you must explicitly set the RNG seed for each chain 2 3 3 Samplers A Sampler is an object that acts on a set of parameters and updates them from one iteration to the next During initialization of the model Samplers are chosen automatically for all parameters The Model holds an internal list of Sampler Factory objects which inspect the graph recognize sets of parameters that can be updated with specific methods and generate Sampler objects for them The list of Sampler Factories is traversed in order starting with sampling
27. m zero sin x Sine Real sinh x Hyperbolic Sine Real sqrt x Square root Real z gt 0 step x Test for z gt 0 Logical tan x Tangent Real tanh x Hyperbolic Tangent Real trunc x Round to integer Integer towards zero 24 Table 5 2 Scalar functions in the bugs module Distribution Density Distribution Quantile Bernoulli dbern pbern qbern Beta dbeta pbeta qbeta Binomial dbin pbin qbin Chi square dchisqr pchisqr qchisqr Double exponential ddexp pdexp qdexp Exponential dexp pexp qexp F df pf qf Gamma dgamma pgamma qgamma Generalized gamma dgen gamma pgen gamma qgen gamma Noncentral hypergeometric dhyper phyper qhyper Logistic dlogis plogis qlogis Log normal dlnorm plnorm qlnorm Negative binomial dnegbin pnegbin qnegbin Noncentral Chi square dnchisqr puchisqr qnchisqr Normal dnorm pnorm qnorm Pareto dpar ppar qpar Poisson dpois ppois qpois Student t dt pt qt Weibull dweib pweib qweib Table 5 3 Functions to calculate the probability density probability function and quantiles of some of the distributions provided by the bugs module Link function Description Range Inverse cloglog y lt x Complementary log log 0 lt y lt 1 y lt icloglog x log y lt x Log 0 lt y y lt exp x logit y lt x Logit O lt y lt 1 y lt ilogit x probit y lt x Probit O lt y lt 1 y lt phi x Table 5 4 Link functions in the bugs module 25 Function Description Restrictions inprod x1 x2 Inn
28. meter values are valid for their distribution 6 1 Distribution aliases A distribution may optionally have an alias which can be used in the model definition in place of the canonical name Aliases are used to to avoid confusion with other statistical software in which distributions may have different names Table 6 4 shows the distributions in the bugs module with an alias 28 Name Usage Density Lower Upper Beta dbeta a b qe py 0 1 a gt 0 b gt 0 B a b Chi square dchisqr k goat exp 2 2 0 ee 21 5 Double ddexp mu tau pale Uli exponential 7 gt 0 E ie Exponential dexp lambda 0 iSi Aexp Ax F df n m r 2t Ron 3 i 0 n gt 0 m gt 0 TEE mm 7 Urmi Gamma dgamma r lambda dx exp Az 0 A gt 0 r gt 0 I r Generalized dgen gamma r lambda b bd 1 expf Ax 0 gamma A gt 0 b gt 0 r gt 0 I r Logistic dlogis mu tau Texp 1 u r T gt 0 1 exp x u r Log normal dlnorm mu tau ait 0 SS i 72x 1 exp r log x 2 Noncentral dnchisqr k delta yo exp 3 _ a 2 r D exp 2 0 Chi squre k gt 0 6 gt 0 r 0 y 2 24 D Er Normal dnorm mu tau 1 T gt 0 37 exp 2 p 7 Pareto dpar alpha c act e c a gt 0 c gt 0 Student t dt mu tau k rm 4 2 1 4s Hei y 5 T gt 0 k gt 0 TE Mer k Uniform dunif a b 1 a b a lt b b a Weibull dweib v lambda jai or 0 20 0 vir exp Ax Table 6 1 Univariate re
29. monitor that calculates an estimate of the effective number of parameters on the model 11 monitor clear lt varname gt type lt montype gt Clears the monitor of the given type associated with variable lt varname gt 14 3 1 9 CODA coda lt varname gt stem lt filename gt If the named node has a trace monitor this dumps the monitored values of to files CODAindex txt CODAindex1 out CODAindex2 txt in a form that can be read by the coda package of R The stem option may be used to modify the prefix from CODA to another string The wild card character may be used to dump all monitored nodes 3 1 10 EXIT exit Exits JAGS JAGS will also exit when it reads an end of file character 3 1 11 DATA TO data to lt filename gt Writes the data i e the values of the observed nodes to a file in the R dump format The same file can be used in a DATA IN statement for a subsequent model See also DATA IN 3 1 2 3 1 12 PARAMETERS TO parameters to lt file gt chain lt n gt Writes the current parameter values i e the values of the unobserved stochastic nodes in chain lt n gt to a file in R dump format The name and current state of the RNG for chain lt n gt is also dumped to the file The same file can be used as input ina PARAMETERS IN statement in a subsequent run See also PARAMETERS IN 3 1 4 3 1 13 SAMPLERS TO samplers to lt file gt Writes out a summary of t
30. mpletely adapted but it may be preferable to run the model again with a longer adaptation phase starting from the MODEL IN statement Alternatively you may use an ADAPT statement see below immediately after initialization 3 1 7 ADAPT adapt lt n gt by lt m gt Updates the model by n iterations keeping the model in adaptive mode and prints a message to indicate whether adaptation is successful Successive calls to ADAPT may be made until the adaptation is successful The next call to UPDATE then turns off adaptive mode immediately Use this instead of the first UPDATE statement if you want explicit control over the length of the adaptive sampling phase Like the UPDATE statement the ADAPT statement prints a progress bar but with plus signs instead of asterisks 3 1 8 MONITOR In JAGS a monitor is an object that calculates summary statistics from a model The most commonly used monitor simply records the value of a single node at each iteration This is called a trace monitor monitor lt varname gt thin n type lt montype gt The thin option sets the thinning interval of the monitor so that it will only record every nth value The thin option selects the type of monitor to create The default type is trace More complex monitors can be defined that do additional calculations For example the dic module defines a deviance monitor that records the deviance of the model at each iteration and a pD
31. nderscores an important fact only the stochastic nodes in a model are really important Deterministic nodes are merely a syntacti cally convenient way of describing the relations between or transformations of the stochastic nodes 2 1 2 Data The data are defined in a separate file from the model definition in the format created by the dump function in R The simplest way to prepare your data is to read them into R and then dump them Only numeric vectors matrices and arrays are allowed More complex data structures such as factors lists and data frames cannot be parsed by JAGS nor can non numeric vectors Any R attributes of the data such as names and dimnames are stripped when they are read into JAGS The data may contain missing values but you cannot supply partially missing values for a multivariate node In JAGS a node is either completely observed or completely unobserved The unobserved nodes are referred to as the parameters of the model The data file therefore defines the parameters of the model by omission Here are the data for the LINE example xf lt c 1 2 3 4 5 R style comments like this one can be embedded in the data file cyt lt c 1 3 3 3 5 N lt 5 It is an error to supply a data value for a deterministic node See however section 7 0 3 on observable functions 2 1 3 Node Array dimensions Array declarations JAGS allows the option of declaring the dimensions of node arra
32. om effects in the same block Currently the methods only work on parameters that have a normal prior distribution Some of the samplers are based in the idea of introducing latent normal variables that reduce the GLM to a linear model This idea was introduced by Albert and Chib 1 for probit regression with a binary outcome and was later refined and extended to logistic regression with binary outcomes by Holmes and Held 7 Another approach auxiliary mixture sam pling was developed by Frihwirth Schnatter et al 5 and is used for more general Poisson regression and logistic regression models with binomial outcomes Gamerman 6 proposed a stochastic version of the iteratively weighted least squares algorithm for GLMs which is also implemented in the glm module However the IWLS sampler tends to break down when there are many random effects in the model It uses Metropolis Hastings updates and the acceptance probability may be very small under these circumstances Block updating in GLMMs frees the user from the need to center predictor variables like this y il dnorm mu i tau mu i lt alpha beta x i mean x The second line can simply be written mu i lt alpha beta x i without affecting the mixing of the Markov chain 21 Chapter 5 Functions Functions allow deterministic nodes to be defined using the lt gets operator Most of the functions in JAGS are scalar functions taking scalar arguments H
33. one in the DATA IN statement The chain option may be omitted in which case the parameter values in all chains are set to the same value The PARAMETERS IN statement must be used after the COMPILE statement and before the INITIALIZE statment You may only supply the values of unobserved stochastic nodes in the parameters file not logical or constant nodes See also PARAMETERS TO 3 1 12 3 1 5 INITIALIZE initialize Initializes the model using the previously supplied data and parameter values supplied for each chain 13 3 1 6 UPDATE update lt n gt by lt m gt Updates the model by n iterations If JAGS is being run interactively a progress bar is printed on the standard output consisting of 50 asterisks If the by option is supplied a new asterisk is printed every m iterations If this entails more than 50 asterisks the progress bar will be wrapped over several lines If m is zero the printing of the progress bar is suppressed If JAGS is being run in batch mode then the progress bar is suppressed by default but you may activate it by supplying the by option with a non zero value of m If the model has an adaptive sampling phase the first UPDATE statement turns off adaptive mode for all samplers in the model after n 2 iterations A warning is printed if adaptation is incomplete Incomplete adaptation means that the mixing of the Markov chain is not optimal It is still possible to continue with a model that has not co
34. ons see the file COPYING for details Chapter 2 Running a model in JAGS JAGS is designed for inference on Bayesian models using Markov Chain Monte Carlo MCMC simulation Running a model refers to generating samples from the posterior distribution of the model parameters This takes place in five steps 1 Definition of the model 2 Compilation 3 Initialization 4 Adaptation and burn in 5 Monitoring The next stages of analysis are done outside of JAGS convergence diagnostics model criticism and summarizing the samples must be done using other packages more suited to this task There are several R packages designed for analyzing MCMC output and JAGS can be used from within R using the rjags package 2 1 Definition There are two parts to the definition of a model in JAGS a description of the model and the definition of the data 2 1 1 Model definition The model is defined in a text file using a dialect of the BUGS language The model definition consists of a series of relations inside a block delimited by curly brackets and and preceded by the keyword model Here is the standard linear regression example model 4 for i in 1 N Y i dnorm mu i tau mu i lt alpha beta x i x bar x bar lt mean x alpha dnorm 0 0 1 0E 4 beta dnorm 0 0 1 0E 4 sigma lt 1 0 sqrt tau tau dgamma 1 0E 3 1 0E 3 Each relation defines a node in the model in terms of other nodes that appear on
35. ori to lie in a certain range Although BUGS has no construct for representing truncated variables it turns out that there is no difference between censoring and truncation for top level parameters i e variables with no unobserved parents Hence for example this theta dnorm 0 1 0E 3 ICO is a perfectly valid way to describe a parameter 0 with a half normal prior distribution Prior ordering occurs when a vector of nodes is known a priori to be strictly increasing or decreasing It can be represented in OpenBUGS with symmetric constructs e g X 1 dnorm 0 1 0E 3 1 X 2 1 X 2 dnorm 0 1 0E 3 I X 1 ensures that X 1 lt X 2 JAGS makes an attempt to separate these three concepts Censoring is handled in JAGS using the new distribution dinterval section 7 0 3 This can be illustrated with a survival analysis example A right censored survival time t with a Weibull distribution is described in OpenBUGS as follows tli dweib r mu i I c il 34 where t is unobserved if t gt c In JAGS this becomes is censored i dinterval t i c il t i dweib r mu i where is censored i is an indicator variable that takes the value 1 if t is censored and 0 otherwise See the MICE and KIDNEY examples in the classic bugs set of examples Truncation is represented in JAGS using the T construct which has the same syntax as the I construct in OpenBUGS but has a different interpretation
36. owever JAGS also allows arbitrary vector and array valued functions such as the matrix multiplication operator and the transpose function t defined in the bugs module and the matrix exponential function mexp defined in the msm module JAGS also uses an enriched dialect of the BUGS language with a number of operators that are used in the S language Scalar functions taking scalar arguments are automatically vectorized They can also be called when the arguments are arrays with conforming dimensions or scalars So for example the scalar c can be added to the matrix A using B lt At c instead of the more verbose form D lt dim A for i in 1 D 1 for j in 1 D 2 B i j lt Ali jl c 5 1 Base functions The functions defined by the base module all appear as infix or prefix operators The syntax of these operators is built into the JAGS parser They are therefore considered part of the modelling language Table 5 1 lists them in reverse order of precedence Logical operators convert numerical arguments to logical values zero arguments are con verted to FALSE and non zero arguments to TRUE Logical and comparison operators return the value 1 if the result is TRUE and 0 if the result is FALSE Comparison operators are non associative the expression x lt y lt z for example is syntactically incorrect The special function is an exception in table 5 1 It is not a function defined by the base module but is a place
37. ows data transformations but the syntax is different from BUGS BUGS allows you to put a stochastic node twice on the left hand side of a relation as in this example taken from the manual for i in 1 N z i lt sqrt y il z i dnorm mu tau This is forbidden in JAGS You must put data transformations in a separate block of relations preceded by the keyword data data for i in 1 N z i lt sqrt y il 32 model 4 for i in 1 N z i dnorm mu tau This syntax preserves the declarative nature of the BUGS language In effect the data block defines a distinct model which describes how the data is generated Each node in this model is forward sampled once and then the node values are read back into the data table The data block is not limited to logical relations but may also include stochastic relations You may therefore use it in simulations generating data from a stochastic model that is different from the one used to analyse the data in the model statement This example shows a simple location scale problem in which the true values of the parameters mu and tau are generated from a given prior in the data block and the generated data is analyzed in the model block data for i in 1 N y i dnorm mu true tau true mu true dnorm 0 1 tau true dgamma 1 3 model 4 for i in 1 N y i dnorm mu tau mu dnorm 0 1 0E 3 tau dgamma 1 0E 3 1 0E 3
38. r The popt monitor works exactly like the mean monitor for pp but records contributions to the optimism of the expected deviance Popt The total optimism pop gt Popt can be added to the mean deviance to give the penalized expected deviance 10 The mean monitor for Popt is created with monitor popt type mean Its values can be written out to a file POPTtable0 txt with coda popt type mean step POPT 20 Under asymptotically favourable conditions in which pp lt 1V1 Popt Y 2pp For generalized linear models a better approximation is n PD Port EY yo ager PD The popt monitor uses importance weights to estimate Popit The resulting estimates may be numerically unstable when pp is not small This typically occurs in random effects models so it is recommended to use caution with the popt until I can find a better way of estimating Popt 4 5 The msm module The msm module defines the matrix exponential function mexp and the multi state distribution dmstate which describes the transitions between observed states in continuous time multi state Markov transition models 4 6 The glm module The glm module implements samplers for efficient updating of generalized linear mixed mod els The fundamental idea is to do block updating of the parameters in the linear predictor The glm module is built on top of the Csparse and CHOLMOD sparse matrix libraries 4 3 which allows updating of both fixed and rand
39. s that do not pool values over iterations is to create an index file and one or more output files The index file is has three columns with one each line 10 Pool Pool Output files iterations chains no no CODAindex txt CODAchain1 txt CODAchainN txt no yes CODAindex0 txt CODAchain0 txt yes no CODAtablel txt CODAtableN txt yes yes CODAtable0 txt Table 2 1 Output files created by the CODA command depending on whether a monitor pools its values over chains or over iterations 1 A string giving the name of the scalar value being recorded 2 The first line in the output file s 3 The last line in the output file s The output file s contain two columns 1 The iteration number 2 The value at that iteration Some monitors pool values over iterations For example a mean monitor may record only the sample mean of a node without keeping the individual values from each iteration Such monitors are written out to a table file with two columns 1 A string giving the name of the scalar value being recorded 2 The value pooled over all iterations 11 Chapter 3 Running JAGS JAGS has a command line interface To invoke jags interactively simply type jags at the shell prompt on Unix or the Windows command prompt on Windows To invoke JAGS with a script file type jags lt script file gt A typical script file has the following commands model in line bug Read model file data in line data
40. s the number of elements in a node array and the dim function returns a vector containing the dimensions of an array These two functions may be used to simplify the data preparation For example if Y represents a vector of observed values then using the length O function in a for loop for i in 1 length y Y i dnorm mu i tau avoids the need to put a separate data value N in the file representing the length of Y For multi dimensional arrays the dim function serves a similar purpose The dim function returns a vector which must be stored in an array before its elements can be accessed For this reason calls to the dim function must always be in a data block see section 7 0 4 data D lt dim Z model for i in 1 D 11 4 for j in 1 D 2 Z i j lt dnorm alpha il betal j tau Clearly the length and dim functions can only work if the size of the node array can be inferred using one of the three methods outlined above Note the length and dim functions are different from all other functions in JAGS they do not act on nodes but only on node arrays As a consequence an expression such as dim a b is syntactically incorrect 2 2 Compilation When a model is compiled a graph representing the model is created in computer memory Compilation can fail for a number of reasons 1 The graph contains a directed cycle These are forbidden in JAGS 2 A top level parameter is undef
41. tically at start time Others may be loaded by the user 4 1 The base module The base module supply the base functionality for the JAGS library to function correctly It is loaded first by default 4 1 1 Base Samplers The base module defines samplers that use highly generic update methods These sampling methods only require basic information about the stochastic nodes they sample Conversely they may not be fully efficient Three samplers are currently defined 1 The Finite sampler can sample a discrete valued node with fixed support of less than 20 possible values The node must not be bounded using the T construct 2 The Real Slice Sampler can sample any scalar real valued stochastic node 3 The Discrete Slice Sampler can sample any scalar discrete valued stochastic node 18 4 1 2 Base RNGs The base module defines four RNGs taken directly from R with the following names 1 base Wichmann Hil1 2 base Marsaglia Multicarry 3 base Super Duper 4 base Mersenne Twister A single RNG factory object is also defined by the base module which will supply these RNGs for chains 1 to 4 respectively if RNG name is not specified in the initial values file All chains generated by the base RNG factory are initialized using the current time stamp If you have more than four parallel chains then the base module will recycle the same for RNGs but using different seeds If you want many parallel chains then you may
42. vector or matrix valued functions in the bugs module The sort and rank functions behaves like their R namesakes sort accepts a vector and returns the same values sorted in ascending order rank returns a vector of ranks This is distinct from OpenBUGS which has two scalar valued functions rank and ranked 5 3 Function aliases A function may optionally have an alias which can be used in the model definition in place of the canonical name Aliases are used to to avoid confusion with other software in which functions may have different names Table 5 7 shows the functions in the bugs module with an alias 26 Function Canonical Alias Compatibile name with Arc cosine arccos acos R Hyperbolic arc cosine arccosh acosh R Arc sine arcsin asin R Hyperbolic arc sine arcsinh asinh R Arc tangent arctan atan R Table 5 7 Functions with aliases in bugs module 27 Chapter 6 Distributions Distributions are used to define stochastic nodes using the operator The distributions defined in the bugs module are listed in table 6 1 real valued distributions 6 2 discrete valued distributions and 6 3 multivariate distributions Some distributions have restrictions on the valid parameter values and these are indicated in the tables If a Distribution is given invalid parameter values when evaluating the log likelihood it returns oo When a model is initialized all stochastic nodes are checked to ensure that the initial para
43. wish to load the lecuyer module 4 1 3 Base Monitors The base module defines the TraceMonitor class type trace This is the monitor class that simply records the current value of the node at each iteration 4 2 The bugs module The bugs module defines some of the functions and distributions from OpenBUGS These are described in more detail in sections 5 and 6 The bugs module also defines conjugate samplers for efficient Gibbs sampling 4 3 The mix module The mix module defines a novel distribution dnormmix mu tau pi representing a finite mix ture of normal distributions In the parameterization of the dnormmix distribution y 7 and Tr are vectors of the same length and the density of y dnormmix mu tau pi is L Fuln 7 T Dnr p r y m where is the probability density function of a standard normal distribution The mix module also defines a sampler that is designed to act on finite normal mixtures It uses tempered transitions to jump between distant modes of the multi modal posterior distribution generated by such models 8 2 The tempered transition method is computa tionally very expensive If you want to use the dnormmix distribution but do not care about label switching then you can disable the tempered transition sampler with set factory mix TemperedMix off type sampler 19 4 4 The dic module The dic module defines new monitor classes for Bayesian model criticism using deviance base
44. ys in the model file The declarations consist of the keyword var for variable followed by a comma separated list of array names with their dimensions in square brackets The dimensions may be given in terms of any expression of the data that returns a single integer value In the linear regression example the model block could be preceded by var x N Y N mu N alpha beta tau sigma x bar Undeclared nodes If a node array is not declared then JAGS has three methods of determining its size 1 Using the data The dimension of an undeclared node array may be inferred if it is supplied in the data file 2 Using the left hand side of the relations The maximal index values on the left hand side of a relation are taken to be the dimensions of the node array For example in this case for i in 1 N for j in 1 M Y i j dnorm mu i j tau Y would be inferred to be an N x M matrix This method cannot be used when there are empty indices e g Y i on the left hand side of the relation 3 Using the dimensions of the parents If a whole node array appears on the left hand side of a relation then its dimensions can be inferred from the dimensions of the nodes on the right hand side For example if A is known to be an N x N matrix and B lt inverse A Then B is also an N x N matrix Querying array dimensions The JAGS compiler has two built in functions for querying array sizes The length function return
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