If MetropolisHastings is the only sampler available for the specified model (see Table 37.8) or if the METROPOLIS option is specified in the BAYES statement, PROC FMM uses the MetropolisHastings approach of Gamerman (1997). See the section Metropolis and MetropolisHastings Algorithms for a general discussion of the MetropolisHastings algorithm.
The Gamerman (1997) algorithm derives a specific density that is used to generate proposals for the componentspecific parameters . The form of this proposal density is multivariate normal, with mean and covariance matrix derived as follows.
Suppose is the vector of model coefficients in the jth component and suppose that has prior distribution . Consider a generalized linear model (GLM) with link function and variance function . The pseudoresponse and weight in the GLM for a weighted least squares step are




If the model contains offsets or FREQ or WEIGHT statements, or if a trials variable is involved, suitable adjustments are made to these quantities.
In each component, , form an adjusted crossproduct matrix with a “pseudo” border
where is a diagonal matrix formed from the pseudoweights w, is a vector of pseudoresponses, and c is arbitrary. This is basically a system of normal equations with ridging, and the degree of ridging is governed by the precision and mean of the normal prior distribution of the coefficients. Sweeping on the leading partition leads to




where the generalized inverse is a reflexive, inverse (see the section Linear Model Theory of Chapter 3: Introduction to Statistical Modeling with SAS/STAT Software, for details).
PROC FMM then generates a proposed parameter vector from the resulting multivariate normal distribution, and then accepts or rejects this proposal according to the appropriate MetropolisHastings thresholds.