Asymptotic Enumeration of Binary Contingency Tables and Comparison with Independence Heuristic

TL;DR

This paper derives precise asymptotic counts for certain high-dimensional binary contingency tables with non-uniform margins and shows that the classical independence heuristic significantly overestimates their number, especially as dimensions grow.

Contribution

It provides a sharp asymptotic formula for counting binary contingency tables with non-uniform margins and compares it to the independence heuristic, revealing overestimation factors.

Findings

Asymptotic formula for contingency tables with non-uniform margins

Independence heuristic overestimates counts by exponential factor

Explicit bounds for correlation ratio constants

Abstract

For parameters $n,\delta,B,C$, we obtained a sharp asymptotic formula for the number of $(n+\lfloor n^\delta\rfloor)^2$-dimensional binary contingency tables with non-uniform margins taking values of $\lfloor BCn\rfloor$ and $\lfloor Cn\rfloor$. Furthermore, we compared our sharp asymptotics with the classical independence heuristic estimate and proved that the independence heuristic overestimates by a factor of $e^{\Theta(n^{2\delta})}$. Our comparison is based on the analysis of the correlation ratio and an explicit bound for the constant in $\Theta$ is also obtained.