The MCMC Procedure
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Overview
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Getting Started
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Syntax
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Details
How PROC MCMC Works Blocking of Parameters Sampling Methods Tuning the Proposal Distribution Conjugate Sampling Initial Values of the Markov Chains Assignments of Parameters Standard Distributions Usage of Multivariate Distributions Specifying a New Distribution Using Density Functions in the Programming Statements Truncation and Censoring Some Useful SAS Functions Matrix Functions in PROC MCMC Create Design Matrix Modeling Joint Likelihood Regenerating Diagnostics Plots Caterpillar Plot Posterior Predictive Distribution Handling of Missing Data Floating Point Errors and Overflows Handling Error Messages Computational Resources Displayed Output ODS Table Names ODS Graphics
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Examples
Simulating Samples From a Known Density Box-Cox Transformation Logistic Regression Model with a Diffuse Prior Logistic Regression Model with Jeffreys’ Prior Poisson Regression Nonlinear Poisson Regression Models Logistic Regression Random-Effects Model Nonlinear Poisson Regression Random-Effects Model Multivariate Normal Random-Effects Model Change Point Models Exponential and Weibull Survival Analysis Time Independent Cox Model Time Dependent Cox Model Piecewise Exponential Frailty Model Normal Regression with Interval Censoring Constrained Analysis Implement a New Sampling Algorithm Using a Transformation to Improve Mixing Gelman-Rubin Diagnostics
- References
| Sampling Methods |
When possible, PROC MCMC uses conjugate sampling algorithms on the parameters (see the section Conjugate Sampling). If conjugacy is not attainable, PROC MCMC samples according to the Table 54.5. Each block of parameters is classified by the nature of the prior distributions. “Continuous” means all priors of the parameters in the same block have a continuous distribution. “Discrete” means all priors are discrete. “Mixed” means that some parameters are continuous and others are discrete. Parameters that have binary priors are treated differently, as indicated in the table. MVN stands for the multivariate normal distribution, and MVT stands for the multivariate t distribution.
Blocks |
Default Method |
Alternative Method |
|---|---|---|
Continuous |
MVN |
MVT |
Discrete (other than binary) |
Binned MVN |
Binned MVT or symmetric geometric |
Mixed |
MVN |
MVT |
Binary (single dimensional) |
Inverse CDF |
|
Binary (multidimensional) |
Independence sampler |
|
Random effect |
Normal |
For a block of continuous parameters, PROC MCMC uses a multivariate normal distribution as the default proposal distribution. In the tuning phase, the procedure finds an optimal scale 
and a tuning covariance matrix 
.
For a discrete block of parameters, PROC MCMC uses a discretized multivariate normal distribution as the default proposal distribution. The scale and covariance matrix 
are tuned. Alternatively, you can use an independent symmetric geometric proposal distribution. The density has form 
and has variance 
. In the tuning phase, the procedure finds an optimal proposal probability 
for every parameter in the block.
You can change the proposal distribution, from the normal to a t distribution. You can either use the PROC option PROPDIST=T(df) or PARMS statement option </ T(df)> to make the change. The t distributions have thicker tails, and they can propose to the tail areas more efficiently than the normal distribution. It can help with the mixing of the Markov chain if some of the parameters have a skewed tails. See Nonlinear Poisson Regression Models. The independence sampler (see the section Independence Sampler) is used for a block of binary parameters. The inverse CDF method is used for a block that consists of a single binary parameter.
For univariate random effects, PROC MCMC uses a normal density random walk Metropolis algorithm on each of the random-effects parameters. If the random effect is multivariate, then a random walk Metropolis based on a multivariate normal proposal distribution is used. There is no alternative sampling method avaliable for random-effects parameters.