Sampling contingency tables

Abstract

Let Ω be the set of all m × n matrices, where r i and c j are the sums of entries in row i and column j, respectively. Sampling efficiently uniformly at random elements of Ω is a problem with interesting applications in Combinatorics and Statistics. To calibrate the statistic χ 2 for testing independence, Diaconis and Gangolli proposed a Markov chain on Ω that samples uniformly at random contingency tables of fixed row and column sums. Although the scheme works well for practical purposes, no formal proof is available on its rate of convergence. By using a canonical path argument, we prove that this Markov chain is fast mixing and the mixing time τ x ( ϵ ) is given by τ x ( ϵ ) ≤ 2 c m a x ( m n ) 4 ln ( c m a x ϵ − 1 ) ,where c m a x − 1 is the maximal value in a cell. Keywords: Sampling, Canonical path, Mixing time of Markov chain