Random Transpositions on Contingency Tables
Abstract
Contingency tables are useful objects in statistics for representing 2way data. With fixed row and column sums, and a total of $n$ entries, contingency tables correspond to parabolic double cosets of $S_n$. The uniform distribution on $S_n$ induces the FisherYates distribution, classical for its use in the chisquared test for independence. A Markov chain on $S_n$ can then induce a random process on the space of contingency tables through the double cosets correspondence. The random transpositions Markov chain on $S_n$ induces a natural `swap' Markov chain on the space of contingency tables; the stationary distribution of the Markov chain is the FisherYates distribution. This paper describes this Markov chain and shows that the eigenfunctions are orthogonal polynomials of the FisherYates distribution. Results for the mixing time are discussed, as well as connections with sampling from the uniform distribution on contingency tables, and data analysis.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 DOI:
 10.48550/arXiv.2208.10700
 arXiv:
 arXiv:2208.10700
 Bibcode:
 2022arXiv220810700S
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability
 EPrint:
 39 pages, 1 figure