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

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 $n\ll N$. Individuals 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.