bayesian output analysis program (boa) version 1.1 user`s manual
Contents
1. FILE MENU 3 Import Data gt gt 4 Load Session 5 Save Session 6 Exit BOA rs ee E Selection 3 1 Import Data Menu BOA can import MCMC output from a variety of sources Data may be added to the analysis via the import menu at any point in the analysis Three common data formats are supported IMPORT DATA MENU Li sera atest ssss ssa 3 CODA Output Files 4 Flat ASCII File 5 Data Matrix Object 6 View Format Specifications 7 Options 8 Selection 3 1 1 Data Options The Options menu item lists the values for the user settings used to import data 10 Data Parameters 1 Working Directory 2 ASCII File Ext txt Select parameter to change or press lt ENTER gt to continue 1 Most users will want to specify the Working Directory at the start of their BOA session This directory should be set to the path in which the MCMC output files are stored The specified working directory should not be terminated with a slash 3 1 2 CODA Output Files The two CODA output files generated by the Bayesian inference Using Gibbs Sam pling BUGS or WinBUGS program can be imported into BOA The output file containing the parameter definitions should be saved as a ind file whereas the file containing the sampler output should be saved as a out file BOA will expect these files to be located in the Working Directory See Section 3 1 1 for instructions on specifying the working directory Upon choosing to2. 13 If set to TRUE titles are added to the plots otherwise a value of FALSE will suppress plot titles 14 If set to TRUE all plots generated in BOA will be kept open otherwise a value of FALSE indicates that only the most recently opened plots be kept open 15 The number of rows and columns respectively of plots to display in one graphics window 16 If set to TRUE only one chain is displayed per plot otherwise a value of FALSE forces all of the chains to be displayed on the same plot 36 Chapter 7 Options Menu The Options Menu serves as a central location from which the options in Sections 3 1 1 5 3 and 6 3 can be accessed GLOBAL OPTIONS MENU 3 Analysis 4 Data 5 Plot 6 A11 NTRA Selection 37 Chapter 8 Window Menu The Window Menu allows the user to switch between and save the active graphics windows WINDOW 2 MENU 3 Previous 4 Next 5 Save to Postscript File 6 Close 7 Close All Di ASP SS Selection The number of the active graphics window is displayed in the title of his menu In this example graphics window 1 is the active window 8 1 Previous Graphics Window Make the previous graphics window in the list of open windows the active graphics window 8 2 Next Graphics Window Make the next graphics window in the list of open windows the active graphics win dow 38 8 3 Save to Postscript File Saves the ac
3. New parameter name none 1 sigma Read 1 items Define the new parameter as a function of the parameters listed above 1 1 sqrt tau Read 1 items sigma has now been added to the two datasets in BOA and will be available to all subsequent analyses 4 3 Display Master Dataset Selecting this option will display summary information for the Master dataset MMARY INFORMATIO Iterations Min Max Sample linet 1 200 200 line2 1 200 200 Support linet alpha beta tau sigma Min Inf Inf 0 0 Max Inf Inf Inf Inf Support line2 alpha beta tau sigma Min Inf Inf 0 0 Max Inf Inf Inf Inf 4 4 Reset The Reset option copies the Master dataset to the Working dataset This undoes any modifications that were made to the Working dataset 18 Chapter 5 Analysis Menu The statistical analysis procedures are accessible through the Analysis Menu Analy ses are categorized into two groups Descriptive Statistics and Convergence Diag nostics ANALYSIS MENU a ag O 3 Descriptive Statistics gt gt 4 Convergence Diagnostics gt gt 5 Options A se Selection 5 1 Descriptive Statistics Menu Options to compute autocorrelations cross correlations and summary statistics are available from the Descriptive Statistics Menu DESCRIPTIVE STATISTICS MENU De Se es Se ee ee 3 Autocorrelations 4 Correlation Matrix 5 Highest Probability Density Intervals 6 Summary Statistics Ya
4. Rubin Convergence Diagnostic The code for implementing the Gelman and Rubin 1992 convergence diagnostic in BOA is based on the itsim function contributed to the Statlib archive by Andrew Gelman http lib stat cmu edu The Brooks Gelman and Rubin convergence diagnostic is appropriate for the analysis of two or more parallel chains each with different starting values which are overdispersed with respect to the target distribution Several methods for generating starting values for the MCMC samplers have been proposed Gelman and Rubin 1992 Applegate et al 1990 Jennison 1993 The following diagnostic information was obtained for the line example BROOKS GELMAN AND RUBIN CONVERGENCE DIAGNOSTICS Iterations used 101 200 Potential Scale Reduction Factors alpha beta tau 0 9962501 1 0019511 1 0099913 Multivariate Potential Scale Reduction Factor 1 010112 Corrected Scale Reduction Factors Estimate 0 975 alpha 1 107170 1 116686 beta 1 087270 1 131090 tau 1 027212 1 090423 The diagnostic originally proposed by Gelman and Rubin 1992 is based on a com parison of the within and between chain variance for each variable This comparison is used to estimate the potential scale reduction factor PSRF the multiplicative factor by which the sampling based estimate of the scale parameter of the marginal posterior distribution might be reduced if the chains were run to infinity To adjust for the sampling variability in the
5. chain contains only those parameters common to all chains CAUTION Although possible to do so convergence diagnostics and autocorrela tions should not be computed for combined chains A combined chain is essentially a single chain with potentially multiple samples per iteration These analyses expect that a single chain has no more than one sample per iteration 4 1 2 Delete Chain Chains may be discarded when they are no longer needed Discarding chains may free up a substantial amount of computer memory The program prompts the user to select the chain s to discard DELETE CHAINS linei line2 Specify chain index or vector of indices none 1 The specified chain s will be immediately deleted from the Master dataset If the Working dataset has not been modified the chain s will be deleted from there as well If modifications were made to the Working dataset the user can copy the new Master dataset to the Working dataset via the Reset option If no chain is entered at the prompt no action is taken 4 1 3 Subset Chains Subsets of the MCMC sequences can be selected for analysis via the Subset option SUBSET WORKING DATASET Specify the indices of the items to be included in the subset Alternatively items may be excluded by supplying negative indices Selections should be in the form of a number or numeric vector Chains linei line2 Specify chain indices all 1 c 1 2 Parameters 15 1 2 3 alpha b
6. that were kept the number of iterations that were discarded and the Cramer von Mises statistic Failure of the chain to pass this test indicates that a longer run of the MCMC sampler is needed in order to achieve convergence A halfwidth test is performed on the portion of the chain that passes the sta tionarity test for each variable Spectral density estimation is used to compute the 24 asymptotic standard error of the mean If the halfwidth of the confidence interval for the mean is less than a specified fraction accuracy of this mean the halfwidth test indicates that the mean is estimated with acceptable accuracy The confidence level and accuracy can be modified through Options 5 and 6 respectively in Section 5 3 Failure of the halfwidth test implies that a longer run of the MCMC sampler is needed to increase the accuracy of the estimated posterior mean 5 2 4 Raftery and Lewis Convergence Diagnostic The Raftery and Lewis convergence diagnostic is appropriate for the analysis of indi vidual chains The following diagnostic information was obtained for the line example RAFTERY AND LEWIS CONVERGENCE DIAGNOSTIC Quantile 0 025 Accuracy 0 02 Probability 0 9 Chain linet Thin Burn in Total Lower Bound Dependence Factor alpha 1 2 160 165 0 969697 beta 1 5 188 165 1 139394 sigma d 2 160 165 0 969697 The diagnostic proposed by Raftery and Lewis 1992b tests for convergence to the stationary distribution and es
7. variance estimates the correction proposed by Brooks 22 and Gelman 1998 is applied to the PSRF to produce the corrected scale reduction factor CSRF BOA also displays an upper quantile of the sampling distribution for the CSRF Users can control which quantile is computed via Option 1 in Section 5 3 Brooks and Gelman 1998 developed a multivariate extension to the PSRF known as the multivariate potential scale reduction factor MPSRF The MPSRF does not include a correction for sampling variability This statistic is relevant when interest lies in general multivariate functionals of the chain The MPSRF and the PSRF satisfy the following relationship maz PSRF lt MPSRF Computation of the reduction factors is based on analysis of variance and sampling from the normal distribution To avoid violations of the latter assumption BOA transforms any parameters specified to be restricted to the range a b to the loga rithmic or logit scale before calculating this diagnostic By default only the second half of the chains iterations 101 200 is used to compute the reduction factors Op tion 2 in 5 3 can be used to vary the proportion of samples from the end of the chains to be included in the analysis If the estimates are approximately equal to one or as a rule of thumb the 0 975 quantile is lt 1 2 the samples may be considered to have arisen from the stationary distribution In this case descriptive statistics may be calculated fo
8. BAYESIAN OUTPUT ANALYSIS PROGRAM BOA VERSION 1 1 USER S MANUAL Brian J Smith January 8 2005 Contents 1 Getting Started 4 dd SOpiaimne BOA 2 3 0 alee a oe Be Oe Bee PR eS he 4 1 2 Win BGS Line Example 00 ble bens Sh gs aay ae eg ad 5 1 2 1 Bayesian Model hasi a a a Soares 5 T22 Win BUGS Code es ade E A A A da e 5 1 23 Saving the WinBUGS Sampler Output 6 E AAA gk eo Be em Bie T 2 Using the BOA Menu Driven User Interface 8 3 File Menu 10 g amp l Import Data Men io pra da O 2 Se PORN A Se eee oe te o 10 dul Data OPti ns so LA Se Oe BSG ee eh BES 10 aie CODA Output Blesa ro a se PEA SEES EE eS 11 dla FBGA CU Eile sta ds eet Be a Aa a 11 3 1 4 Data Matrix Object Ls 12 3 1 5 View Format Specifications 03 ear e 12 3 2a AAO BG o SION A E T E 12 Bio Save Segs e A ele tog ahd A a ee a pee dd 12 Sr BBO A a ee eh ee Gee Bd 13 4 Data Management Menu 14 41 Chains Ment soe stea ate do ea ls A ayes ar 14 4 1 1 Combine All Chains oy wane a e a a 14 4 1 2 Delete Ch in a ds By hap ra mag a he ea alea i 15 ADL Subset GRAINS doae e e we Re AD ed SG i 15 4 2 Parameters Menu dass amp ALS a te ae at doth di dt es 16 4 2 1 Set Parameter Bounds on 324 ae Rias Bele eee da od 16 4 2 2 Delete Parameters 0 0 0 0 00 0000 o 4 2 3 Create New Parameters 0 0 0000088 4 5 Display Master Dataset es CA ee ee be Se et la JA ROSCOE st A A IN A Me Rs aAa TN Analysis Menu 5 1 Descriptive S
9. E pls as aoe A E a Or NA A SA Se a ee ee eo AN a l T T T T T T T T 0 50 100 150 0 50 100 150 First Iteration in Segment First Iteration in Segment g s D o D o a Q T 7 Lt ek SS O DO ee ee ee TA A ee O A E ES T T T T l T T T T 0 50 100 150 0 50 100 150 First Iteration in Segment First Iteration in Segment N F Qo SSS SS E Se a GSS SSS SA J line1 line2 o Bs oe 3179 o ad es o o o Y o 2 ball BO ibs eat E ee ce oy o o T T T T T T T T 0 50 100 150 0 50 100 150 First Iteration in Segment First Iteration in Segment 39 6 3 Plot Options Plot Parameters 1 Number of Bins 20 2 Window Fraction 0 5 Density 3 Bandwidth function x 0 5 diff range x log length x 1 4 Kernel gaussian Gelman amp Rubin 5 Alpha Level 0 05 6 Number of Bins 20 7 Window Fraction 0 5 Geweke 8 Alpha Level 0 05 9 Number of Bins 10 10 Window 1 Fraction 0 1 11 Window 2 Fraction 0 5 Graphics 12 Legend TRUE 13 Title TRUE 14 Keep Previous Plots FALSE 15 Plot Layout c 3 2 16 Plot Chains Separately FALSE Select parameter to change or press lt ENTER gt to continue The options grouped under the Graphics heading control the general layout used to generate plots The following gives a brief description of each of these options 12 If set to TRUE legends are included in the plots otherwise a value of FALSE will suppress plot legends
10. aaa a ha Tal a n Selection 5 1 1 Autocorrelations This option produces lag autocorrelations for the monitored parameters within each chain High autocorrelations indicate slow mixing within a chain and usually slow convergence to the posterior distribution 19 LAGS AND AUTOCORRELATIONS Lag 1 Lag 5 Lag 10 Lag 50 alpha 0 10005297 0 04361973 0 001152681 0 06391649 beta 0 07166133 0 10149584 0 059398063 0 07936142 tau 0 32327917 0 06211792 0 064798232 0 01946111 sigma 0 42629373 0 11736382 0 103620199 0 11424204 Option 11 in Section 5 3 allows the user to set the lags at which autocorrelations are computed 5 1 2 Correlation Matrix This option returns the correlation matrix for the parameters in each chain High correlation among parameters may lead to slow convergence to the posterior As sociated models may need to be reparameterized in order to reduce the amount of cross correlation CROSS CORRELATION MATRIX alpha beta tau sigma alpha 1 beta 0 1643217 1 tau 0 0556438 0 0416541 1 sigma 0 0937184 0 0422862 0 66123 1 5 1 3 Highest Probability Density Intervals Highest probability density HPD interval estimation is one means of generating Bayesian posterior intervals HPD intervals span a region of values containing 1 a x 100 of the posterior density such that the posterior density within the interval is always greater than that outside Consequently HPD intervals are of the short est length of an
11. actor and the maximum of the potential scale reduction factors see Section 5 2 1 for successively larger segments of the chains The first segment contains the first 50 iterations in the chains The remaining iterations are then partitioned into equal bins and added incrementally to construct the remaining segments Option 1 in Section 6 3 governs the number of bins used for the plot Scale factors are plotted against the maximum iteration number in the segments Cubic splines are used to interpolate through the point estimates from the segments Brooks amp Gelman Multivariate Shrink Factors A deu En oO N A 4 Rmax E i 1 1 j LO 1 wn g O 2 Oo E o E lt 179 LO o 4 a o o 4 A 50 100 150 200 Last Iteration in Segment 33 6 2 2 Gelman and Rubin Plot Plots the Gelman and Rubin corrected potential scale reduction factors see Section 5 2 1 for each parameter in successively larger segments of the chain The first segment contains the first 50 iterations in the chain The remaining iterations are then partitioned into equal bins and added incrementally to construct the remaining segments Options 5 and 6 in Section 6 3 control the error rate for the upper quantile and the number of bins respectively Option 7 determines the proportion of samples from the end of the chains to be included in the analysis The scale factor is plotted against the maximum iteration number for the segme
12. acy 6 Alpha Level Raftery amp Lewis 7 Accuracy 8 Alpha Level 9 Delta 10 Quantile Statistics 11 ACF Lags 12 Alpha Level 13 Batch Size 14 Quantiles 005 05 001 025 oooo c 1 5 10 50 0 05 50 c 0 025 0 5 0 975 26 Chapter 6 Plot Menu Like the Analysis Menu the Plot Menu categorizes the available plots into a De scriptive and Convergence Diagnostic group Most of the options found under the Analysis Menu have a counterpart within the Plot Menu PLOT MENU 3 Descriptive gt gt 4 Convergence Diagnostics gt gt 5 Options 6 Selection 6 1 Descriptive Plot Menu DESCRIPTIVE PLOT MENU LS se So4e 3 Autocorrelations 4 Density 5 Running Mean 6 Trace Selection 27 6 1 1 Autocorrelations Plot Plot the first several lag autocorrelations for each parameter in each chain alpha beta tau 0 0 05 1 0 1 0 0 0 0 5 1 0 1 0 00 05 1 0 Sampler Lag Autocorrelations line1 alpha beta tau 28 0 0 05 1 0 1 0 0 0 0 5 1 0 1 0 0 0 05 1 0 1 0 Lag line2 6 1 2 Density Plot Plot the kernel density estimate for each parameters in each chain Options 3 and 4 in Section 6 3 control the width and type of window used in the computations respectively Estimated Posterior Density a wo _ linet foo line2 o co gt g g
13. ain Monte Carlo MCMC sampler The first chain linel was generated with the initial values of alpha 5 beta 5 tau 5 whereas the second chain line2 was generated with alpha 0 01 beta 0 01 tau 0 01 1 2 2 WinBUGS Code The code for the Line Example is given below The WinBUGS seed was set to 12345 after loading the initial values Model main for i in 1 N yli dnorm mu il tau mu i lt alpha beta x i mean x alpha dnorm 0 0 0001 beta dnorm 0 0 0001 tau dgamma 0 001 0 001 Data list N 5 x c 1 2 3 4 5 y c 1 3 3 3 5 Initial values for first chain list tau 5 alpha 5 beta 5 Initial values for second chain list tau 0 01 alpha 0 01 beta 0 01 1 2 3 Saving the WinBUGS Sampler Output In the Sampler Monitor Tool dialog box alpha beta and tau were first specified as the nodes Then the Update Tool dialog box was used to generate two hundred MCMC samples for each of the two parallel chains BOA will import sampler output saved in the CODA file format CODA output can be generated by entering an asterisk in the Sample Monitor Tool node list box and pressing the coda button Two windows will appear a window with the sampler output and another with the names of the nodes that were monitored The files should be saved as text files with extensions out and ind respectively Follow the steps below to ensu
14. cting to import a matrix object the user will be asked to Enter object name none 1 linei BOA will import the data from the linel object in the current S PLUS or R session 3 1 5 View Format Specifications Selecting this menu item will display the format specifications for the three types of data that BOA can import CODA CODA output files produced by WinBUGS ind and out files must be located in the Working Directory see Options ASCII ASCII file txt containing the monitored parameters from one run of the sampler file must be located in the Working Directory see Options parameters are stored in space or tab delimited columns parameter names must appear in the first row iteration numbers may be specified in a column labeled iter Matrix Object S or R numeric matrix whose columns contain the monitored parameters from one run of the sampler iteration numbers and parameter names may be specified in the dimnames 3 2 Load Session The Load Session menu item allows users to load previously saved work Enter name of object to load none 1 line 3 3 Save Session All imported data and user settings may be saved at any point in the analysis Users will be prompted to Enter name of object to which to save the session data none 1 line The session data will be saved to the specified S object 12 3 4 Exit BOA Select this item to exit from the BOA program Users will be prompt
15. ed to verify their intention to exit in order to avoid an unintended termination of the program Do you really want to EXIT y n n 1 Users wishing to save their work should go back and do so before exiting BOA will not automatically save the user s work 13 Chapter 4 Data Management Menu BOA offers a wide range of options for managing the imported data Two copies of the data are maintained by the program the Master dataset and the Working dataset The Master dataset is a static copy of the data as it was first imported This copy remains essentially unchanged throughout the BOA session The Working dataset is a dynamic copy that can be modified by the user All analyses are performed on the Working dataset The Data Management menu offers the following options DATA MANAGEMENT MENU Li aa aa a a eel ae 3 Chains gt gt 4 Parameters gt gt 5 Display Working Dataset 6 Display Master Dataset Too 813 os Seis i cones Selection 4 1 Chains Menu CHAINS MENU Lara gt 3 Combine All 4 Delete 5 Subset 63452 Selection 4 1 1 Combine All Chains Selecting this options will combine together all of the chains in the Working dataset Sequencing is preserved by concatenating together the different chains and then or 14 dering the result by the iteration numbers in the original chains Note that this may result in a chain with multiple samples at a given iteration The resulting
16. eta tau Specify parameter indices all 1 2 Iterations HA AAA AAA Min Max Sample linet 1 200 200 line2 1 200 200 Specify iterations all 1 50 200 In this example both chains were first included in the subset Since the default is to include all chains this line could have been left blank Next the beta parame ter is excluded by supplying a negative sign in front of the selection Finally the subset is limited to iteration 50 200 Users can verify that the subset was success fully constructed by selecting the option to display the Working dataset output not shown Thinning Thinning refers to the practice of including every kt iteration from a chain Users can thin a chain by using the seq function when prompted to specify the iterations For example the following input will included every other iteration from the chain seq 1 200 length 100 A description of the seg function can be found at the end of the Appendix 4 2 Parameters Menu PARAMETERS MENU DE AN 3 Set Bounds 4 Delete 5 New 63 3 gt Selection 4 2 1 Set Parameter Bounds This option allows the user to specify the lower and upper bounds support of selected MCMC parameters The parameter support factors into the computation of the Brooks Gelman 4 Rubin convergence diagnostics 16 SET PARAMETER BOUNDS linei line2 Specify chain index or vector of indices all 1 Parameters alpha beta tau Spec
17. f the N 0 1 suggest that the chain in the first window had not fully converged The two sided p value outputted by BOA gives the tail probability associated with the observed Z statistic It is common practice to conclude that there is evidence against convergence when the p value is less than 0 05 Otherwise it can be said that the results of this test do not provide any evidence against convergence This does not however prove that the chain has converged 5 2 3 Heidelberger and Welch Convergence Diagnostic The Heidelberger and Welch convergence diagnostic is appropriate for the analysis of individual chains The following diagnostic information was obtained for the line example HEIDLEBERGER AND WELCH STATIONARITY AND INTERVAL HALFWIDTH TESTS Halfwidth test accuracy 0 1 Chain linet Stationarity Test Keep Discard C von M Halfwidth Test Mean alpha passed 200 0 0 037295593 passed 3 0214700 beta passed 200 0 0 008893071 failed 0 8120946 tau passed 200 0 0 126287673 failed 1 9402362 Halfwidth alpha 0 1421230 beta 0 1405493 tau 0 3044104 Heidelberger and Welch s 1983 stationarity test is based on Brownian bridge theory and uses the Cramer von Mises statistic If there is evidence of non stationarity the test is repeated after discarding the first 10 of the iterations This process continues until the resulting chain passes the test or more than 50 of the iterations have been discarded BOA reports the number of iterations
18. g the accuracy of sampling based approaches to calculating posterior moments In Bayesian Statistics 4 eds J M Bernardo J O Berger A P Dawid and A F M Smith Oxford Oxford University Press Heidelberger P and Welch P 1983 Simulation run length control in the presence of an initial transient Operations Research 31 1109 1144 Jennison C 1993 Discussion of Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods by Smith and Roberts Journal of the Royal Statistical Society Series B 55 54 56 Raftery A L and Lewis S 1992a Comment One long run with diagnostics implementation strategies for Markov chain Monte Carlo Statistical Science 7 493 497 42 10 Raftery A L and Lewis S 1992b How many iterations in the Gibbs sampler In Bayesian Statistics 4 eds J M Bernardo J O Berger A P Dawid and A F M Smith Oxford Oxford University Press 43
19. ify parameter index or vector of indices al1 1 3 Specify lower and upper bounds as a vector Inf Inf 1 c 0 Inf In this example the variance parameter tau has been restricted to only non negative values When no chain s is specified the default is to apply the change to all of the chains Likewise the default is to select all parameters and to set the bounds to 8 00 4 2 2 Delete Parameters Often times it may be desired to delete parameters that are not of interest in the analysis This may arise in cases where data other than model parameters were saved to the output file imported into BOA Alternatively the user may only be interested in functions of the original parameters Once the new parameter is created using the methods described in the following section the unnecessary parameter upon which it is based may be deleted Deleted parameters will speed up the manipulation of data in BOA DELETE PARAMETERS alpha beta tau Specify parameter index or vector of indices none 1 4 2 3 Create New Parameters BOA includes an option to create new parameters Most S functions can be used to create the new parameter Typically a new parameter is defined as a function of the 17 existing parameters For instance suppose the user was interested in analyzing the standard deviation sigma 1 ytau The following menu commands demonstrate how to create this new parameter NEW PARAMETER 1 alpha beta tau
20. import CODA output the user will be prompted to Enter filename prefix without the ind out extension Working Directory d bjsmith boa 1 linet Only the filename prefix should be specified BOA will automatically add the appro priate extensions and load the data from the linel ind and line1 out files 3 1 3 Flat ASCII File BOA includes an import filter for general ASCII files This is particularly useful for output generated by custom MCMC programs The ASCII file should contain one run of the sampler with the monitored parameters stored in space or tab delimited columns and with the parameter names in the first row Iteration numbers may be specified in a column labeled iter The ASCII file should be located in the Working Directory Upon selected to import an ASCII file the program will prompt the user to Enter filename prefix without the txt extension Working Directory d bjsmith boa 1 linel Specify only the filename prefix The import filter will automatically add the exten sion and load the data from the linel txt file See Section 3 1 1 for instructions on specifying the Working Directory and the default ASCII file extension 11 3 1 4 Data Matrix Object MCMC output stored as an S object may be imported into BOA The object must be a numeric matrix whose columns contain the monitored parameters from one run of the sampler The iteration numbers and parameter names may be specified in the dimnames Upon sele
21. is program is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 2 of the License or any later version This program is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details For a copy of the GNU General Public License write to the Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA or visit their web site at http www gnu org copyleft gpl html NOTE if the menu unexpectedly terminates type boa menu recover TRUE to restart and recover your work BOA MAIN MENU FAI IC IK 1 File gt gt 2 Data gt gt 3 Analysis gt gt 4 Plot gt gt 5 Options gt gt 6 Window gt gt Selection Note the message given at startup if the menu unexpectedly terminates type boa menu recover TRUE to restart and recover your work There are a few 8 instances where supplying the wrong type of data will crash the menu system Im mediately doing a recover will ensure that no data is lost Chapter 3 File Menu Selecting menu item 1 from the BOA Main Menu brings up the File Menu Options to import data load previously saved session data save the current session and exit the program are available from the File Menu
22. nt Cubic splines are used to interpolate through the point estimates from the segments Gelman amp Rubin Shrink Factors a gt A A 6 Py 97 5 on 97 5 a it Median e Median o Als i TE g 24 Y 7 i ce y A N y J 2 2 T T T T a T T T 50 100 150 200 50 100 150 200 Last Iteration in Segment Last Iteration in Segment N 1 i 97 5 gt y Median ae ets E a D 1 4 1 Y Q 50 100 150 200 Last Iteration in Segment 34 6 2 3 Geweke Plot Plots the Geweke Z statistic see Section 6 3 for each parameter in successively smaller segments of the chain The k segment contains the last number of bins k 1 number of bins 100 of the iterations in the chain Options 8 and 9 in Section 6 3 set the error rate for the confidence bounds and the number of bins included in the plot respectively Options 10 and 11 control the fraction of iterations covered by the windows used in computing the Geweke diagnostic It may be possible that some of the subsets contain too few iterations to compute the test statistic Such segments if they exist are automatically omitted from the plot The test statistic is plotted against the minimum iteration number for the segment Geweke Convergence Diagnostic A SS SS SS eS Se oe ee F o line1 a a a ii tine z i s7 E o 2 o ro oO 5 e o e S o Es o o o o p gt 7 4 l SE ERA E
23. r the combined latter 50 of iterations from all of the chains 5 2 2 Geweke Convergence Diagnostic The Geweke convergence diagnostic is appropriate for the analysis of individual chains when convergence of the mean of some function of the sampled parameters is of interest The following diagnostic information was obtained for the line example GEWEKE CONVERGENCE DIAGNOSTIC Fraction in first window 0 1 Fraction in last window 0 5 Chain linet alpha beta tau Z Score 0 1226878 0 1306432 0 9944621 p value 0 9023544 0 8960575 0 3199980 The chain is divided into two windows containing a set fraction of the first and the last iterations Options 3 and 4 in Section 5 3 allow the user to set the fraction of iterations included in the first and the last window respectively Geweke 1992 proposed a method to compare the mean of the sampled values in the first window 23 to the mean of the sampled values in the last window There should be a sufficient number of iterations between the two windows to reasonably assume that the two means are approximately independent His method produces a Z statistic calculated as the difference between the two means divided by the asymptotic standard error of their difference where the variance is determined by spectral density estimation As the number of iterations approaches infinity the Z statistic approaches the N 0 1 if the chain has converged Z values which fall in the extreme tails o
24. re that WinBUGS saves your CODA files correctly 1 Select the window containing the CODA data to be saved 2 Choose File gt Save As from the WinBUGS menu bar to bring up the Save As dialog box 3 Select Plain Text txt as the Save as type 4 Enter the file name enclosed in quotation marks e g linel out linel ind line2 out line2 ind 5 Specify the directory in which to save the file 6 Press the Save button to complete the save If quotation marks are not used when entering the file names Microsoft Windows will automatically append txt extensions to the file names when saved Carefully follow the previous steps to avoid import problems in BOA that are a result of CODA file names with the incorrect extensions 1 2 4 R Line Data The sampler output from the Line Example is included in the R package To load the data type gt data line at the R command line Two R data matrices linel and line2 will be loaded These may be imported directly into BOA see Section 3 1 4 Chapter 2 Using the BOA Menu Driven User Interface A menu driven interface is supplied with the BOA It provides easy access to all of the command line function To start the menu system type gt boa menu to bring up the main menu Bayesian Output Analysis Program BOA Version 1 1 3 for i386 mingw32 Copyright c 2004 Brian J Smith lt brian j smith uiowa edu gt Th
25. sics 9 1 Output Display Options curra 242 Bod aM Bag tk et a Glee 0027 Vectors ih S s A 35 BS LE Sa he eke ion dene BS ee RU ee Ee ee ds Chapter 1 Getting Started 1 1 Obtaining BOA BOA is available in library format for R and the Microsoft Windows version of S PLUS The R library is available from CRAN at http www r project org It can be downloaded and installed automatically by entering the following at the R command line gt install packages boa The S PLUS library is available from http www public health uiowa edu boa To install extract the BOA zip file to the library directory located in the path where S PLUS is installed Once the appropriate files are installed on your computer type gt library boa at the R or S PLUS command line to load the BOA library 1 2 WinBUGS Line Example 1 2 1 Bayesian Model Output from the BUGS Line example is used to illustrate the capabilities of the BOA program The Line example involves a liner regression analysis of the data points 1 1 2 3 3 3 4 3 and 5 5 The proposed Bayesian model is yli N mauli tau mult alpha beta x x i mean x with the following priors alpha N 0 0 0001 beta N 0 0 0001 tau Gamma 0 001 0 001 Interest lies in estimating the posterior distribution of alpha beta and sigma 1 vtau The starting values for the parameters were varied to generate two parallel chains from the Markov ch
26. t o a 2 a 7 5 5 lt Q S Q m a ome y o i i o o 1 z S 4 4 A estimmins A T T T T 5 0 5 10 beta Y o a o gt O E N 3 o S y 2 o tau 29 6 1 3 Running Mean Plot Generate a time series plot of the running mean for each parameter in each chain The running mean is computed as the mean of all sampled values up to and including that at a given iteration Sampler Running Mean FT A ia a ad A i linet poa line2 o line2 oO The N bo 1 E i a E A I 5 7 TA ale Fo ala S 2 u d Ti lo MN Es s T T T T T T T T T T T 0 50 100 150 200 0 50 100 150 200 Iteration Iteration i Y m S a 4 1 A I olla T T T T T 0 50 100 150 200 Iteration 30 6 1 4 Trace Plot Generate a time series plot of the sampled points for each parameter in each chain Sampler Trace o 2 linet line2 l atl N z I 1 T Q y ie 1 o 4 i i i if F We I a l lo SST 1 T T T T T T T T T 50 100 150 200 0 50 100 150 200 Iteration Iteration N o foe i L o 4 Iteration 31 6 2 Convergence Diagnostics Plot Menu CONVERGENCE DIAGNOSTICS PLOT MENU QoS Sa Se ae 3 Brooks amp Gelman 4 Gelman amp Rubin 5 Geweke l a Selection 32 6 2 1 Brooks and Gelman Plot Plots the Brooks and Gelman multivariate potential scale reduction f
27. tatistics Menu 36 0 amp eee ae BE ke Be 5 1 1 Autocorrelations Awa Se do it o i Sethe deat AE Ay 5 1 2 Correlation Matrix ac a A A o hl 5 1 3 Highest Probability Density Intervals Sido Summary o palstlo sos apa S e A A ek 5 2 Convergence Diagnostics Menu eee 5 2 1 Brooks Gelman Rubin Convergence Diagnostic 5 2 2 Geweke Convergence Diagnostic 5 2 3 Heidelberger and Welch Convergence Diagnostic 5 2 4 Raftery and Lewis Convergence Diagnostic AA gs Sahoo Gig ee BE O Plot Menu 6 1 Descriptive Plot Menu LS EA E A E 6 1 1 Autocorrelations Plot 61 2 Density Plot persai dr A Ae id ide 6 1 3 Running Mean Plot 3 9 4 amp oh ee A See ed 6 1 4 Trace Plots or daa ad RS te o 6 2 Convergence Diagnostics Plot Menu 6 2 1 Brooks and Gelman Plot 6 2 2 Gelman and Rubin Plot 6 2 3 Geweke Plot 0000000000008 6 3 Plot Options rrea a OF es eee de a Gee A Options Menu Window Menu 8 1 Previous Graphics Window e e 8 2 Next Graphics Window 2 00 ab LE E a da 8 3 Save to Postscript File dd ct e a a o de 8 4 Close Graphics Window de bis De De tr Ea 8 5 Close All Graphics Window 63 0 ae SAA AA AAA 19 19 19 20 20 21 Al 22 23 24 25 26 27 27 28 29 30 31 32 33 34 35 36 37 9 S PLUS and R Ba
28. timates the run lengths needed to accurately estimate quantiles of functions of the parameters The user may specify the quantile of interest the desired degree of accuracy in estimating this quantile and the probability of attaining the indicated degree of accuracy Options 7 9 and 10 in Section 5 3 allow the user to modify these quantities BOA computes the lower bound the number of iterations needed to estimate the specified quantile to the desired accuracy using independent samples If fewer iterations than this bound have been loaded into BOA the following warning is displayed seem Warning 0k Available chain length is 200 Re run simulation for at least 3746 iterations OR reduce the quantile accuracy or probability to be estimated If sufficient MCMC iterations are available BOA lists the lower bound the total number of iterations needed for each parameter the number of initial iterations to discard as the burn in set and the thinning interval to be used The dependence factor measures the multiplicative increase in the number of iterations needed to reach convergence due to within chain correlation Dependence factors greater than 5 0 often indicate convergence failure and a need to reparameterize the model Raftery and Lewis 1992a 25 5 3 Analysis Options Analysis Parameters 1 Alpha Level 2 Window Fraction Geweke 3 Window 1 Fraction 4 Window 2 Fraction Heidelberger Welch 5 Accur
29. tive graphics window to a postscript file The user is prompted to enter the name of the postscript file in which to save the contents of the graphics window Enter name of file to which to save the plot none 1 Only the name of the file should be given The file will be automatically saved in the Working Directory see Section 3 1 1 Microsoft Windows users may save the graphics window in other formats directly from the S PLUS or R program menus 8 4 Close Graphics Window Close the active graphics window 8 5 Close All Graphics Window Closes all open graphics windows 39 Chapter 9 S PLUS and R Basics 9 1 Output Display Options The options function in S PLUS and R can be used to control the format of the outputted text in BOA This should be done prior to starting BOA To set the number of significant digits to be displayed type options digits lt value gt The number of characters allowed per line can be controlled by entering the command options width lt value gt 9 2 Vectors in S Several menu selections in BOA prompt the user to input a vector of data Vectors in S can be supplied in a variety of ways The simplest way to construct a vector is with the concatenation function c c lt element 1 gt lt element 2 gt lt element n gt where the elements may be numerical or logical values or character strings Another means of constructing vectors is with the seg function seq lt starting val
30. ue gt lt ending value gt length lt number of values gt or 40 seq lt starting value gt lt ending value gt by lt step size gt where length is number of values in the vector and by is the spacing between successive values in the vector The operator which is a special case of the seq function can also be used to construct vectors This operator can be defined as lt starting value gt lt ending value gt seq lt starting value gt lt ending value gt by 1 For more detailed information about these functions consult the help systems in S PLUS or R 41 Bibliography 1 Applegate D Kannan R and Polson N G 1990 Random polynomial time algorithms for sampling from joint distributions Technical report no 500 Carnegie Mellon University Brooks S and Gelman A 1998 General methods for monitoring convergence of iterative simulations Journal of Computational and Graphical Statistics 7 4 434 455 Brooks S P and Roberts G O 1998 Convergence assessment techniques for Markov chain Monte Carlo Statistics and Computing 8 4 319 335 Cowles M K and Carlin B P 1996 Markov chain Monte Carlo convergence diagnostics a comparative review Journal of the American Statistical Associa tion 91 883 904 Gelman A and Rubin D B 1992 Inference from iterative simulation using multiple sequences Statistical Science 7 457 511 Geweke J 1992 Evaluatin
31. ut the table SUMMARY STATISTICS Batch size for calculating Batch SE and Lag 1 ACF 50 Chain linet Mean SD Naive SE MC Error Batch SE Batch ACF 0 025 alpha 3 0214700 0 5210029 0 03684047 0 07251309 0 04842256 0 7384625 2 0480500 beta 0 8120946 0 3519652 0 02488770 0 07171012 0 01329908 0 7084603 0 2435375 tau 9402362 1 8348540 0 12974377 0 15531429 0 18201157 0 3526486 0 2042925 sigma 0 9987152 0 5574588 0 03941829 0 07653543 0 06009981 0 2221603 0 3932961 0 5 0 975 MinIter MaxIter Sample alpha 3 0115000 4 378725 1 200 200 beta 0 7870000 1 555925 1 200 200 tau 1 3480000 6 465950 1 200 200 sigma 0 8613953 2 214427 pl 200 200 e Options 13 and 14 in Section 5 3 allow the user to change the batch size and the quantiles respectively See the Appendix for instructions on setting the number of significant digits and display width 5 2 Convergence Diagnostics Menu The Convergence Diagnostics Menu offers the user the following diagnostic methods CONVERGENCE DIAGNOSTICS MENU 21 3 Brooks Gelman amp Rubin 4 Geweke 5 Heidelberger amp Welch 6 Raftery amp Lewis 7 Selection These are the most commonly used methods used to asses the convergence of MCMC output A brief explanation of each approach is given in the following sections Users are referred to the work of Brooks and Roberts 1998 and Cowles and Carlin 1996 for a more in depth review and comparison of these methods 5 2 1 Brooks Gelman amp
32. y of the Bayesian intervals The algorithm described by Chen and Shao 1999 is used to compute the HPD intervals in BOA under the assumption of unimodal marginal posterior distributions The alpha level for the intervals can be modified through Option 12 in Section 5 3 HIGHEST PROBABILITY DENSITY INTERVALS Alpha level 0 05 Chain linet Lower Bound Upper Bound 20 alpha 1 9470000 3 937000 beta 0 1762000 1 491000 tau 0 0618500 5 767000 sigma 0 3347497 2 074796 5 1 4 Summary Statistic This option prints summary statistics for the parameters in each chain The sample mean and standard deviation are given in the first two columns These are followed by three separate estimates of the standard error 1 a naive estimate the sample standard deviation divided by the square root of the sample size which assumes the sampled values are independent 2 a time series estimate the square root of the spectral density variance estimate divided by the sample size which gives the as ymptotic standard error Geweke 1992 and 3 a batch estimate calculated as the sample standard deviation of the means from consecutive batches of size 50 divided by the square root of the number of batches The autocorrelation between batch means follows and should be close to zero If not the batch size should be increased Quan tiles are given after the batch autocorrelation Finally the minimum and maximum iteration numbers and the total sample size round o
Download Pdf Manuals
Related Search