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# Bayesian Analysis Procedures

• FMM — Finite mixture models
• GENMOD — Generalized linear models
• LIFEREG — Parametric models for failure time data that can be uncensored, right censored, left censored, or interval censored
• MCMC — General purpose Markov chain Monte Carlo (MCMC) simulation procedure that is designed to fit Bayesian models with arbitrary priors and likelihood functions
• PHREG — Regression analysis of survival data based on the Cox proportional hazards model

# Bayesian Analysis

Bayesian methods treat parameters as random variables and define probability as "degrees of belief" (that is, the probability of an event is the degree to which you believe the event is true). It follows from these postulates that probabilities are subjective and that you can make probability statements about parameters. When performing a Bayesian analysis you begin with a prior belief regarding the probability distribution of an unknown parameter. After learning information from observed data, you change or update your belief about the unknown parameter and obtain a posterior distribution. In theory, Bayesian methods offer simple alternatives to statistical inferenceâ€”all inferences follow from the posterior distribution. In practice, however, you can obtain the posterior distribution with straightforward analytical solutions only in the most rudimentary problems. Most Bayesian analyses require sophisticated computations, including the use of simulation methods. You generate samples from the posterior distribution and use these samples to estimate the quantities of interest.

Below are highlights of the capabilities of the SAS/STAT procedures that perform Bayesian analysis:

• a general purpose Markov chain Monte Carlo (MCMC) simulation procedure that fits Bayesian models with arbitrary priors and likelihood functions
• Bayesian modeling and inference for the following types of models:
• generalized linear models
• finite mixture models
• accelerated life failure models
• Cox regression (semi-parametric proportional hazards model)
• piecewise exponential models