# Bayesian Analysis Using SAS/STAT Software

The use of Bayesian methods has become increasingly popular in modern statistical analysis, with applications in a wide variety of scientific fields. Bayesian methods incorporate existing information (based on expert knowledge, past studies, and so on) into your current data analysis. This existing information is represented by a prior distribution, and the data likelihood is effectively weighted by the prior distribution as the data analysis results are computed. The main outcomes of a Bayesian analysis are the posterior distributions of a model's parameters, rather than point estimates and their standard errors. Access to a model's parameters' posterior distributions enables you to address scientific questions of interest directly, because once the model parameters are estimated, it is easy to compute the posterior distributions for any functions of the parameters or any quantities of interest. Some statisticians produce Bayesian analyses simply to operate in the Bayesian framework. They use noninformative priors that produce very similar results to, say, maximum likelihood-based analysis.

For an overview of Bayesian statistics, see the following:

#### SAS/STAT software surfaces Bayesian methods in three ways:

Bayesian Counterparts to Standard Analyses Available through Existing Procedures

The FMM, GENMOD, LIFEREG, and PHREG procedures offer convenient access to Bayesian analysis for finite mixture models, generalized linear models, accelerated life failure models, Cox regression models, and piecewise exponential hazard models. In each of these procedures, you specify your model essentially the same way you do for a frequentist approach, but you add a BAYES statement to request Bayesian estimation methods for fitting the model. The BAYES statement requests that the parameters of the model be estimated by Markov chain Monte Carlo sampling techniques and provides options that enable you to specify prior information, control the sampling, obtain posterior summary statistics and convergence diagnostics, and save the posterior samples to a SAS data set for further analysis. For sampling from posterior distributions, adaptive rejection sampling (ARS), conjugate sampling, Metropolis, and Gamerman algorithms can be specified for different models.

Specialized Models with Wide Application

The BGLIMM and BCHOICE procedures are high-performance procedures that are tailored to fit generalized linear mixed models (GLMMs) and discrete choice models, respectively. These two procedures perform Bayesian estimation only, and therefore they do not require a BAYES statement.

PROC BGLIMM fits GLMMs similar to the types of model used in PROC MIXED and PROC GLIMMIX. The procedure adopts familiar SAS syntax in specifying GLMMs and employs efficient Markov chain Monte Carlo (MCMC) sampling tools to estimate the posterior distributions of model parameters.

PROC BCHOICE performs Bayesian analysis for discrete choice models, which are used in marketing research and microeconomics to model decision makers choices among alternative products and services. PROC BCHOICE fits a number of choice models, including multinomial logit, multinomial probit, and nested logit models.

A General-Purpose Bayesian Modeling Procedure

The MCMC procedure is a flexible, general purpose Markov chain Monte Carlo simulation procedure that fits Bayesian models with arbitrary priors and likelihood functions.