The RANDOM Statement and More: Moving On with PROC MCMC

The MCMC procedure, first released in SAS/STAT® 9.2, provides a flexible environment for fitting a wide range of Bayesian statistical models. Key enhancements in SAS/STAT 9.22 and 9.3 offer additional functionality and improved performance. The RANDOM statement provides a convenient way to specify linear and nonlinear random-effects models along with substantially improved performance. The MCMC procedure also supports multivariate distributions, such as the multivariate normal and inverse-Wishart distributions, and implements conjugate sampling methods when appropriate to improve sampling speed. This paper describes key enhancements of PROC MCMC in SAS/STAT 9.22 and 9.3 and illustrates the use of these enhancements with examples. INTRODUCTION The MCMC procedure is a Bayesian sampling procedure based on Markov chain Monte Carlo methods. First released in SAS/STAT 9.2, PROC MCMC accommodates a broad range of Bayesian statistical models, and its main sampling mechanism is a self-tuned random walk Metropolis algorithm. You can use the MCMC procedure to fit linear or nonlinear models that take various forms (for example, a multilevel model in which the parameters have a nonlinear relationship with the response variable and the likelihood takes on a nonstandard form). For an overview of the MCMC procedure, see Chen (2009). This paper focuses on two key enhancements to PROC MCMC in SAS/STAT 9.3: the RANDOM statement and multivariate distributions. It also illustrates the usage of the PREDDIST statement (available in SAS/STAT 9.22) for posterior prediction and explains the newly implemented conjugate sampling algorithms that are available in SAS/STAT 9.3. USING PROC MCMC To use PROC MCMC, you specify the model parameters, the prior distributions, and the likelihood function (the conditional distribution of the response variable given the parameters and covariates). The prior and likelihood function jointly define a posterior distribution, which becomes the objective function that PROC MCMC uses in simulation. The simplest call to the MCMC procedure has the following form: PROC MCMC options; PARMS; declare model parameters PRIOR; declare prior distributions, log. . // Programming statements; MODEL; declare the likelihood function, log.f .yi j //, for each observation Run; The options control simulation setup, posterior calculation, convergence diagnostics, and plotting. The MCMC procedure enables you to work with either standard distributions (normal, Poisson, truncated gamma, and so on) or construct a general distribution using SAS programming statements. If you are working with a large number of parameters, you can use multiple PARMS statements to place them in blocks. Each block of parameters is updated in the Metropolis algorithm conditionally in every iteration. By default, PROC MCMC assumes that all observations are independent and calculates the posterior density (on the logarithm scale) as log.p. jy// D log. . //C n X iD1 log.f .yi j //C C where can be a multidimensional parameter vector, f .yi j / is the likelihood function for a single observation in the data set, n is the sample size, and C is a constant that can be ignored. 1 Statistics and Data Analysis SAS Global Forum 2011