Learning the distribution of latent variables in paired comparison models with round-robin scheduling

TL;DR

This paper investigates the asymptotic behavior of maximum likelihood estimators for latent ability distributions in paired comparison models with round-robin scheduling, using graphical models and Markov chain techniques.

Contribution

It introduces a graphical model framework for analyzing paired comparison data with round-robin scheduling and derives asymptotic properties and risk bounds for the estimators.

Findings

Proves geometric loss of memory in the graphical model.

Establishes asymptotic behavior of the likelihood function.

Provides risk bounds for the maximum likelihood estimator.

Abstract

Paired comparison data considered in this paper originate from the comparison of a large number N of individuals in couples. The dataset is a collection of results of contests between two individuals when each of them has faced n opponents, where n is much larger than N. Individual are represented by independent and identically distributed random parameters characterizing their abilities.The paper studies the maximum likelihood estimator of the parameters distribution. The analysis relies on the construction of a graphical model encoding conditional dependencies of the observations which are the outcomes of the first n contests each individual is involved in. This graphical model allows to prove geometric loss of memory properties and deduce the asymptotic behavior of the likelihood function. This paper sets the focus on graphical models obtained from round-robin scheduling of these contests.Following a classical construction in learning theory, the asymptotic likelihood is used to measure performance of the maximum likelihood estimator. Risk bounds for this estimator are finally obtained by sub-Gaussian deviation results for Markov chains applied to the graphical model.