What does a random contingency table look like?

TL;DR

This paper studies the structure of random contingency tables with fixed margins, showing they are typically close to a uniquely defined 'typical table' that maximizes a specific concave function.

Contribution

It introduces the concept of a 'typical table' as the unique maximizer of a concave function, characterizing the typical structure of random contingency tables with fixed margins.

Findings

Random contingency tables are close to the typical table with high probability.

The typical table uniquely maximizes a concave function g(X) over the set of tables with fixed margins.

The function g(X) is strictly concave and determines the typical structure of the tables.

Abstract

Let R=(r_1, ..., r_m) and C=(c_1, ..., c_n) be positive integer vectors such that r_1 +... + r_m=c_1 +... + c_n. We consider the set Sigma(R, C) of non-negative mxn integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D in Sigma(R,C) is close with high probability to a particular matrix ("typical table'') Z defined as follows. We let g(x)=(x+1) ln(x+1)-x ln x for non-negative x and let g(X)=sum_ij g(x_ij) for a non-negative matrix X=(x_ij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative mxn matrices X with row sums R and column sums C at a unique point, which we call the typical table Z.