Counting & Sampling Contingency Tables

Dr. M. Cryan
22 April 2009
Suppose we are given two lists r and c of positive integers, where r=(r[1],...., r[m]) represents a list of prescribed row sums and c=(c[1], ..., c[n]) is a list of prescribed column sums. We require that (r[1] + ... + r[m]) =(c[1] + ... + c[n]). In this setting, we say that a m-by-n matrix X of non-negative integers is a Contingency Table (for these given row/column values) if X simultaneously satisfies all of the given row and column sums. The problem of determining whether at least one contingency table exists can be solved in polynomial-time (in fact, this question is fairly trivial).

In my talk, we are interested in the more-difficult problem of randomly sampling a table uniformly at random, from the entire set of contingency tables. This problem has some applications in practical statistics which I will mention. We study a very natural Markov chain on the set of contingency tables called the 2-by-2 heat bath: one step of this chain operates by selecting 2 rows and 2 columns uniformly at random, computing the induced row sums and column sums on this 2-by-2 window, then replacing the window with a table chosen randomly from all 2-by-2 tables with the induced row and column sums. This Markov chain converges to the uniform distribution on contingency tables - our goal is to show that it approaches uniformity within polynomial-time. We are able to achieve this result for the case when the number of rows m is some fixed constant. Our proof is by application of the canonical paths method of Jerrum and Sinclair.

(Joint work with Martin Dyer, Leslie Goldberg, Mark Jerrum and Russell Martin)
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