Commutative Algebra of Statistical Ranking

TL;DR

This paper explores the algebraic structures underlying statistical ranking models, representing orderings as poset chains and analyzing their algebraic varieties, with a focus on toric and non-toric models.

Contribution

It introduces a combinatorial algebraic framework for ranking models, including the analysis of their algebraic varieties and Markov bases, highlighting differences between toric and non-toric models.

Findings

Most models are toric, except the Plackett-Luce model.

The paper characterizes the algebraic varieties of these models.

It examines the Markov bases for the toric models.

Abstract

A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett-Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszar model, the inversion model, and the Bradley-Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis.