Sophisticated Bayesian methods are often used to identify a collection of alleles
that are jointly associated with a particular disease. A disease might not be
expressed when only one of these alleles is present, but each associated allele
might interact with each other in a rather complicated way, causing a disease to
be expressed. In investigating a patient's susceptibility to a disease, it is often
useful to group the collection of associated alleles according to their risk factors.
Our goal is to find the most likely grouping structure of alleles
associated with Rheumatoid Arthritis given a case-control data. The number
of ways to group these m alleles is given by the mth Bell number Bm. For 10 alleles, this
translates to 115,975 groupings. For m = 15, we have over a billion ways to group
the alleles. Clearly computing the probability for each grouping soon become
intractable. A combination of Metropolis-Hastings and local search algorithm is
proposed to accomplish this task. This strategy is first implemented on simulated
data, with a sufficiently large sample size and a known grouping structure, and
the correct grouping is obtained. Stable results are obtained as the algorithm is
run multiple times on Rheumatoid Arthritis data.