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 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. 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.

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 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.

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) 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 the procedure. The default is INF= . For example, PROC MCMC considers 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 .

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:

• create ARRAY symbols to store all data set variables that are used in the program. This can be accomplished with the BEGINCNST and ENDCNST statements.

• program the joint log-likelihood function by using these ARRAY symbols only. The MODEL statement specifies the joint log-likelihood function for the entire data set. Typically, you use the function GENERAL in the MODEL statement.

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.

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 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.

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 requesting plots. 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 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:

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

• PLOTS=(TRACE AUTOCORR) displays the trace and autocorrelation plots for each parameter side by side with two parameters per panel:

  Display 1	Trace(X1)	Autocorr(X1)
  	Trace(X2)	Autocorr(X2)
  Display 2	Trace(X3)	Autocorr(X3)
  	Trace(X4)	Autocorr(X4)

• PLOTS(GROUPBY=TYPE)=(TRACE AUTOCORR) displays all the paneled trace plots, followed by panels of autocorrelation plots:

  Display 1	Trace(X1)
  	Trace(X2)
  Display 2	Trace(X3)
  	Trace(X4)
  Display 3	Autocorr(X1)	Autocorr(X2)
  	Autocorr(X3)	Autocorr(X4)

• PLOTS(UNPACK)=(TRACE AUTOCORR) displays a separate trace plot and a separate correlation plot, parameter by parameter:

  Display 1	Trace(X1)
  Display 2	Autocorr(X1)
  Display 3	Trace(X2)
  Display 4	Autocorr(X2)
  Display 5	Trace(X3)
  Display 6	Autocorr(X3)
  Display 7	Trace(X4)
  Display 8	Autocorr(X4)

• PLOTS(UNPACK GROUPBY=TYPE)=(TRACE AUTOCORR) displays all the separate trace plots followed by the separate autocorrelation plots:

  Display 1	Trace(X1)
  Display 2	Trace(X2)
  Display 3	Trace(X3)
  Display 4	Trace(X4)
  Display 5	Autocorr(X1)
  Display 6	Autocorr(X2)
  Display 7	Autocorr(X3)
  Display 8	Autocorr(X4)

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:

specifies a proposal distribution for the Metropolis algorithm. See the section Metropolis and Metropolis-Hastings Algorithms. 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:

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 the procedure. 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 the procedure. The maximum SINGDEN allowed is .

specifies options for posterior statistics. By default, PROC MCMC computes the posterior mean, standard deviation, quantiles, and two credible intervals: equal-tail and highest posterior density (HPD). Other available statistics include the posterior correlation and covariance. See the section Summary Statistics 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.

specifies the target acceptance rate for the random walk based Metropolis algorithm. See the section Metropolis and Metropolis-Hastings Algorithms. The numeric value must be between and . 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 for more details. The numeric value must be between and . By default, TARGACCEPTI=0.6.

controls the thinning rate of the simulation. PROC MCMC keeps every th 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 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.