Belief propagation for permutations, rankings, and partial orders

TL;DR

This paper introduces a belief propagation algorithm for estimating permutations and rankings from partial data, enabling inference, model selection, and approximation of linear extensions in complex ordering problems.

Contribution

It develops a novel belief propagation method for permutations and partial orders, connecting probabilistic models with efficient inference techniques.

Findings

Derives a belief propagation algorithm for permutation distributions

Provides an approximation method for counting linear extensions

Enables model selection among ranking models

Abstract

Many datasets give partial information about an ordering or ranking by indicating which team won a game, which item a user prefers, or who infected whom. We define a continuous spin system whose Gibbs distribution is the posterior distribution on permutations, given a probabilistic model of these interactions. Using the cavity method we derive a belief propagation algorithm that computes the marginal distribution of each node's position. In addition, the Bethe free energy lets us approximate the number of linear extensions of a partial order and perform model selection between competing probabilistic models, such as the Bradley-Terry-Luce model of noisy comparisons and its cousins.