The MCMC Procedure

specifies a tolerance for acceptance probabilities. By default, ACCEPTTOL=0.075.

specifies the maximum number of autocorrelation lags used in computing the effective sample size; see the section Effective Sample Size in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The value is used in the calculation of the Monte Carlo standard error; see the section Standard Error of the Mean Estimate in Chapter 7: Introduction to Bayesian Analysis Procedures. By default, AUTOCORLAG=MIN(500, MCsample/4), where MCsample is the Markov chain sample size kept after thinning—that is, MCsample . If AUTOCORLAG= is set too low, you might observe significant lags, and the effective sample size cannot be calculated accurately. A WARNING message appears, and you can either increase AUTOCORLAG= or NMC=, accordingly.

specifies the proposal distribution used in sampling discrete parameters. The default is DISCRETE=BINNING.

The keyword values are as follows:

specifies options for MCMC convergence diagnostics. By default, PROC MCMC computes the Geweke test, sample autocorrelations, effective sample sizes, and Monte Carlo errors. The Raftery-Lewis and Heidelberger-Welch tests are also available. See the section Assessing Markov Chain Convergence in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details on convergence diagnostics. You can request all of the diagnostic tests by specifying DIAGNOSTICS=ALL. You can suppress all the tests by specifying DIAGNOSTICS=NONE.

You can use postprocessing autocall macros to calculate convergence diagnostics of the posterior samples after PROC MCMC has exited. See the section Autocall Macros for Postprocessing.

The following options are available.

computes the Deviance Information Criterion (DIC). DIC is calculated using the posterior mean estimates of the parameters. See the section Deviance Information Criterion (DIC) in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details.

specifies the input data set. Observations in this data set are used to compute the log-likelihood function that you specify with PROC MCMC statements.

specifies the numerical definition of infinity in PROC MCMC. The default is INF=1E15. For example, PROC MCMC considers 1E16 to be outside of the support of the normal distribution and assigns a missing value to the log density evaluation. You can select a larger value with the INF= option. The minimum value allowed is 1E10.

specifies options for generating the initial values for the parameters. These options apply only to prior distributions that are recognized by PROC MCMC. See the section Standard Distributions for a list of these distributions. If either of the functions GENERAL or DGENERAL is used, you must supply explicit initial values for the parameters. By default, INIT=MODE. The following keywords are used:

displays the model program and variable lists. The LIST option is a debugging feature and is not normally needed.

displays the compiled program code. The LISTCODE option is a debugging feature and is not normally needed.

specifies how the likelihood function is calculated. By default, PROC MCMC assumes that the observations in the data set are independent so that the joint log-likelihood function is the sum of the individual log-likelihood functions for the observations, where the individual log-likelihood function is specified in the MODEL statement. When your data are not independent, you can specify the JOINTMODEL option to modify the way that PROC MCMC computes the joint log-likelihood function. In this situation, PROC MCMC no longer steps through the input data set to sum the individual log likelihood.

To use this option correctly, you need to do the following two things:

See the sections BEGINCNST/ENDCNST Statement and Modeling Joint Likelihood for details.

specifies an upper limit for the number of proposal tuning loops. By default, MAXTUNE=24. See the section Covariance Tuning for more details.

specifies the maximum number of observation indices to print in the ODS tables “Missing Response Information” table and “Missing Covariates Information” table. This option applies only to programs that model missing data. The default value is 20. MAXINDEXPRINT=ALL prints all observation indices for every missing variable that is modeled in PROC MCMC.

specifies the maximum number of subject values to display in the “Subject Values” column of the ODS table “Random Effects Parameters.” This option applies only to programs that have RANDOM statements. The default value is 20. MAXSUBVALUEPRINT=ALL prints all subject values for every random effect in the program.

controls the display of the Markov chain sampling history.

The default is MCHISTORY=NONE.

specifies a lower limit for the number of proposal tuning loops. By default, MINTUNE=2. See the section Covariance Tuning for more details.

specifies how missing values are handled (see the section Handling of Missing Data for more details). The default is MISSING=COMPLETECASE.

outputs analysis for selected symbols of interest in the program. The symbols can be any of the following: model parameters (symbols in the PARMS statement), secondary parameters (assigned using the operator “=”), the log of the posterior density (LOGPOST), the log of the prior density (LOGPRIOR), the log of the hyperprior density (LOGHYPER) if the HYPER statement is used, or the log of the likelihood function (LOGLIKE). You can use the keyword _PARMS_ as a shorthand for all of the model parameters. PROC MCMC performs only posterior analyses (such as plotting, diagnostics, and summaries) on the symbols selected with the MONITOR= option. You can also choose to monitor an entire array by specifying the name of the array. By default MONITOR=_PARMS_.

Posterior samples of any secondary parameters listed in the MONITOR= option are saved in the OUTPOST= data set. Posterior samples of model parameters are always saved to the OUTPOST= data set, regardless of whether they appear in the MONITOR= option.

specifies the number of burn-in iterations to perform before beginning to save parameter estimate chains. By default, NBI=1000. See the section Burn-in, Thinning, and Markov Chain Samples in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details.

specifies the number of iterations in the main simulation loop. This is the MCMC sample size if THIN=1. By default, NMC=1000.

omits the calculation of the logarithm of the joint distribution of the model parameters at each iteration. The option applies only if all parameters in the model are updated directly from their target distribution, either from the full conditional posterior via conjugacy or from the marginal distribution. Such algorithms do not require the calculation of the joint posterior distribution; hence PROC MCMC runs faster by avoiding these unnecessary calculations. As a result, the OUTPOST= data set does not contain the LOGPRIOR, LOGLIKE, and LOGPOST variables.

specifies the number of threads for simulation. PROC MCMC performs two types of threading. In sampling model parameters, PROC MCMC allocates data to different threads and calculates the objective function by accumulating values from each thread; in sampling of random-effects parameters and missing data variables, each thread generates a subset of these parameters simultaneously at each iteration. Most sampling algorithms, with the exception of conjugate samplers, are threaded. NTHREADS=–1 sets the number of available threads to the number of hyperthreaded cores available on the system. By default, NTHREADS=1.

specifies the number of iterations to use in each proposal tuning phase. By default, NTU=500.

specifies an output data set that contains the posterior samples of all model parameters, the iteration numbers (variable name ITERATION), the log of the posterior density (LOGPOST), the log of the prior density (LOGPRIOR), the log of the hyperprior density (LOGHYPER), if the HYPER statement is used, and the log likelihood (LOGLIKE). Any secondary parameters (assigned using the operator “=”) listed in the MONITOR= option are saved to this data set. By default, no OUTPOST= data set is created.

controls the display of diagnostic plots. Three types of plots can be requested: trace plots, autocorrelation function plots, and kernel density plots. By default, the plots are displayed in panels unless the global plot option UNPACK is specified. Also when more than one type of plot is specified, the plots are grouped by parameter unless the global plot option GROUPBY=TYPE is specified. When you specify only one plot request, you can omit the parentheses around the plot-request, as shown in the following example:

   plots=none
   plots(unpack)=trace
   plots=(trace density)

ODS Graphics must be enabled before plots can be requested. For example:

ods graphics on;
proc mcmc data=exi seed=7 outpost=p1 plots=all;
   parm mu;
   prior mu ~ normal(0, sd=10);
   model y ~ normal(mu, sd=1);
run;
ods graphics off;

For more information about enabling and disabling ODS Graphics, see the section Enabling and Disabling ODS Graphics in Chapter 21: Statistical Graphics Using ODS.

If ODS Graphics is enabled but you do not specify the PLOTS= option, then PROC MCMC produces, for each parameter, a panel that contains the trace plot, the autocorrelation function plot, and the density plot. This is equivalent to specifying PLOTS=(TRACE AUTOCORR DENSITY).

The global-plot-options include the following:

The plot-requests are as follows:

Consider a model with four parameters, X1–X4. Displays for various specifications are depicted as follows.

specifies the method used in constructing the initial covariance matrix for the Metropolis-Hastings algorithm. The QUANEW and NMSIMP methods find numerically approximated covariance matrices at the optimum of the posterior density function with respect to all continuous parameters. The optimization does not apply to discrete parameters. The tuning phase starts at the optimized values; in some problems, this can greatly increase convergence performance. If the approximated covariance matrix is not positive definite, then an identity matrix is used instead. Valid values are as follows:

The optimize-options are as follows:

specifies a proposal distribution for the Metropolis algorithm. See the section Metropolis and Metropolis-Hastings Algorithms in Chapter 7: Introduction to Bayesian Analysis Procedures. You can also use PARMS statement option (see the section PARMS Statement) to change the proposal distribution for a particular block of parameters. Valid values are as follows:

displays the ODS table “Random Effect Observation Information.” The table lists the name of each random effect, the unique values in the corresponding subject variable, the number of observations in each subject, and the observation indices for each subject value.

To understand how this option works, consider the following statements:

data input;
   array names{*} $ n1-n10 ("John" "Mary" "Chris" "Rob" "Greg"
                     "Jen" "Henry" "Alice" "James" "Toby");
   call streaminit(17);
   do i = 1 to 20;
      j = ceil(rand("uniform") * 10 );
      index = names[j];
      output;
   end;
   drop n: j;
run;

proc print data=input;
run;

The input data set (Figure 59.11) contains the index variable, which indicates subjects in a hypothetical random-effects model.

The following statements illustrate the use of the REOBSINFO option:

ods select reobsinfo;
proc mcmc data=input reobsinfo stats=none diag=none;
   random u ~ normal(0, sd=1) subject=index;
   model general(0);
run;

Figure 59.12 displays the “Random Effect Observation Information” table. The table contains the name of the random-effect parameter (u), the values of the subject variable index, the total number of observations, and the row index of these observations in each of the subject values.

The display-options are as follows:

controls the initial multiplicative scale to the covariance matrix of the proposal distribution. By default, SCALE=2.38. See the section Scale Tuning for more details.

specifies the random number seed. By default, SEED=0, and PROC MCMC gets a random number seed from the clock.

controls the number of times that PROC MCMC reports the expected run time of the simulation. This can be useful for monitoring the progress of CPU-intensive programs. For example, with SIMREPORT=2, PROC MCMC reports the simulation progress twice. By default, SIMREPORT=0, and there is no reporting. The expected run times are displayed in the log file.

defines the singularity criterion in PROC MCMC. By default, SINGDEN=1E-11. The value indicates the exclusion of an endpoint in an interval. The mathematical notation “” is equivalent to “” in PROC MCMC—that is, is treated as in PROC MCMC. The maximum SINGDEN allowed is 1E-6.

specifies options for posterior statistics. By default, PROC MCMC computes the posterior mean, standard deviation, quantiles, and two 95% credible intervals: equal-tail and highest posterior density (HPD). Other available statistics include the posterior correlation and covariance. See the section Summary Statistics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. You can request all of the posterior statistics by specifying STATS=ALL. You can suppress all the calculations by specifying STATS=NONE.

You can use postprocessing autocall macros to calculate posterior summary statistics of the posterior samples after PROC MCMC has exited. See the section Autocall Macros for Postprocessing.

The global-stats-options includes the following:

The stats-requests include the following:

specifies the target acceptance rate for the random walk based Metropolis algorithm. See the section Metropolis and Metropolis-Hastings Algorithms in Chapter 7: Introduction to Bayesian Analysis Procedures. The numeric value must be between 0.01 and 0.99. By default, TARGACCEPT=0.45 for models with 1 parameter; TARGACCEPT=0.35 for models with 2, 3, or 4 parameters; and TARGACCEPT=0.234 for models with more than 4 parameters (Roberts, Gelman, and Gilks; 1997; Roberts and Rosenthal; 2001).

specifies the target acceptance rate for the independence sampler algorithm. The independence sampler is used for blocks of binary parameters. See the section Independence Sampler in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The numeric value must be between 0 and 1. By default, TARGACCEPTI=0.6.

controls the thinning rate of the simulation. PROC MCMC keeps every nth simulation sample and discards the rest. All of the posterior statistics and diagnostics are calculated using the thinned samples. By default, THIN=1. See the section Burn-in, Thinning, and Markov Chain Samples in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details.

displays the result of each operation in each statement in the model program as it is executed. This debugging option is very rarely needed, and it produces voluminous output. If you use this option, also use small NMC=, NBI=, MAXTUNE=, and NTU= numbers.

specifies the multiplicative weight used in updating the covariance matrix of the proposal distribution. The numeric value must be between 0 and 1. By default, TUNEWT=0.75. See the section Covariance Tuning for more details.