The MCMC Procedure
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Overview
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Getting Started
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Syntax
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DetailsHow PROC MCMC WorksBlocking of ParametersSampling MethodsTuning the Proposal DistributionDirect SamplingConjugate SamplingInitial Values of the Markov ChainsAssignments of ParametersStandard DistributionsUsage of Multivariate DistributionsSpecifying a New DistributionUsing Density Functions in the Programming StatementsTruncation and CensoringSome Useful SAS FunctionsMatrix Functions in PROC MCMCCreate Design MatrixModeling Joint LikelihoodRegenerating Diagnostics PlotsCaterpillar PlotAutocall Macros for PostprocessingGamma and Inverse-Gamma DistributionsPosterior Predictive DistributionHandling of Missing DataFunctions of Random-Effects ParametersFloating Point Errors and OverflowsHandling Error MessagesComputational ResourcesDisplayed OutputODS Table NamesODS Graphics
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ExamplesSimulating Samples From a Known DensityBox-Cox TransformationLogistic Regression Model with a Diffuse PriorLogistic Regression Model with Jeffreys’ PriorPoisson RegressionNonlinear Poisson Regression ModelsLogistic Regression Random-Effects ModelNonlinear Poisson Regression Multilevel Random-Effects ModelMultivariate Normal Random-Effects ModelMissing at Random AnalysisNonignorably Missing Data (MNAR) AnalysisChange Point ModelsExponential and Weibull Survival AnalysisTime Independent Cox ModelTime Dependent Cox ModelPiecewise Exponential Frailty ModelNormal Regression with Interval CensoringConstrained AnalysisImplement a New Sampling AlgorithmUsing a Transformation to Improve MixingGelman-Rubin Diagnostics
- References
It is impossible to estimate how long it will take for a general Markov chain to converge to its stationary distribution.
It takes a skilled and thoughtful analysis of the chain to decide whether it has converged to the target distribution and
whether the chain is mixing rapidly enough. In some cases, you might be able to estimate how long a particular simulation
might take. The running time of a program that does not have RANDOM statements is approximately linear to the following factors: the number of samples in the input data set, the number of simulations,
the number of blocks in the program, and the speed of your computer. For an analysis that uses a data set of size nsamples, a simulation length of nsim, and a block design of nblocks, PROC MCMC evaluates the log-likelihood function the following number of times, excluding the tuning phase:
![\[ \mbox{\Variable{nsamples}} \times \mbox{\Variable{nsim}} \times \mbox{\Variable{nblocks}} \]](images/statug_mcmc0560.png){kind=small}
The faster your computer evaluates a single log-likelihood function, the faster this program runs. Suppose you have nsamples equal to 200, nsim equal to 55,000, and nblocks equal to 3. PROC MCMC evaluates the log-likelihood function approximately 
times. If your computer can evaluate the log likelihood for one observation 
times per second, this program takes approximately a half a minute to run. If you want to increase the number of simulations
five-fold, the run time increases approximately five-fold.
Each RANDOM statement adds one pass through the input data at each iteration. If the Metropolis algorithm is used to sample the random-effects parameter, the conditional density (objective function) is calculated twice per pass through the data, which requires a computational resource that is approximately equivalent to adding two blocks of parameters.
Of course, larger problems take longer than shorter ones, and if your model is amenable to frequentist treatment, then one of the other SAS procedures might be more suitable. With “regular” likelihoods and a lot of data, the results of standard frequentist analysis are often asymptotically equivalent to a Bayesian approach. If PROC MCMC requires too much CPU time, then perhaps another SAS/STAT tool would be suitable.