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.

Table 54.5 Sampling Methods in PROC MCMC

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.