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The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem. (English) Zbl 0804.62033
Summary: This article describes a method of “grouping” and “collapsing” in using the Gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial. The norms of the forward operators associated with the corresponding nonreversible Markov chains are used to discriminate among different simulation schemes. When applied to Bayesian missing data problems, the idea of collapsing suggests skipping the steps of sampling parameter(s) values in standard data augmentation. By doing this, we obtain a predictive update version of the Gibbs sampler. A procedure of calculating the posterior odds ratio via the collapsed Gibbs sampler when incomplete observations are involved is presented. As an illustration of possible applications, three examples, along with a Bayesian treatment for identifying common protein binding sites in unaligned DNA sequences, are provided.

MSC:
62F15 Bayesian inference
62P10 Applications of statistics to biology and medical sciences; meta analysis
92D20 Protein sequences, DNA sequences
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