# Bayesian Analysis Using the FMM Procedure

The FMM procedure fits statistical models to data for which the distribution of the response is a finite mixture of distributionsthat is, each response is drawn with unknown probability from one of several distributions. You can use PROC FMM to model the component distributions in addition to the mixing probabilities. The procedure is designed to fit finite mixtures of regression models or finite mixtures of generalized linear models in which the covariates and regression structure can be the same across components or might be different.

To perform Bayesian analyses with PROC FMM, you specify a 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, and obtain posterior summary statistics and convergence diagnostics. You can also save the posterior samples to a SAS data set for further analysis.

### Sampling Algorithms and Prior Distributions

The FMM procedure applies highly specialized sampling algorithms in Bayesian models. For single-component models without effects, a conjugate sampling algorithm is used where possible. For models in the exponential family that contain effects, the sampling algorithm is based on Gamerman (1997). For the normal and t distributions, a conjugate sampler is the default sampling algorithm for models with and without effects. In multi-component models, the sampling algorithm is based on latent variable sampling through data augmentation (Frühwirth-Schnatter 2006) and the Gamerman or conjugate sampler. Because of this specialization, the options for controlling the prior distributions of the parameters are limited.

 Effects (exclusive of intercept) Distributions Available Samplers No Normal or t Conjugate or Metropolis-Hastings Yes Normal or t Conjugate or Metropolis-Hastings No Binomial, binary, Poisson, exponential Conjugate or Metropolis-Hastings Yes Binomial, binary, Poisson, exponential Metropolis-Hastings No constant not applicable

The conjugate sampler enjoys greater efficiency than the Metropolis-Hastings sampler and has the advantage of sampling in terms of the natural parameters of the distribution.

You can always switch to the Metropolis-Hastings sampling algorithm in any model by adding the METROPOLIS option in the BAYES statement.