On the number of contingency tables and the independence heuristic

TL;DR

This paper provides precise asymptotic estimates for the number of large contingency tables with specified margins, revealing a phase transition and highlighting limitations of the independence heuristic in certain regimes.

Contribution

It introduces sharp asymptotic formulas for contingency tables with linear margins and identifies a phase transition point affecting heuristic accuracy.

Findings

A second order phase transition at B_c = 1 + sqrt(1 + 1/C)

The independence heuristic undercounts tables for B > B_c

Asymptotic estimates depend on the ratio of margins and reveal critical behavior.

Abstract

We obtain sharp asymptotic estimates on the number of $n \times n$ contingency tables with two linear margins $Cn$ and $BCn$. The results imply a second order phase transition on the number of such contingency tables, with a critical value at \ts $B_{c}:=1 + \sqrt{1+1/C}$. As a consequence, for \ts $B>B_{c}$, we prove that the classical \emph{independence heuristic} leads to a large undercounting.