Bayesian inference of planted matchings: Local posterior approximation and infinite-volume limit

TL;DR

This paper investigates Bayesian inference for matching problems in one dimension, demonstrating local approximation of the posterior and the existence of a well-defined infinite-volume limit for partial matchings, with open questions remaining for higher dimensions.

Contribution

It provides the first analysis of local posterior approximation and infinite-volume limits for Bayesian matchings in one dimension, highlighting differences between exact and partial matchings.

Findings

Posterior for partial matching exhibits decay of correlations as n grows.

Exact matching posterior requires global sorting for local approximation.

Marginal statistics have a well-defined limit with a flow-based indexing in the Poisson process limit.

Abstract

We study Bayesian inference of an unknown matching $\pi^*$ between two correlated random point sets $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$ in $[0,1]^d$, under a critical scaling $\|X_i-Y_{\pi^*(i)}\|_2 \asymp n^{-1/d}$, in both an exact matching model where all points are observed and a partial matching model where a fraction of points may be missing. Restricting to the simplest setting of $d=1$, in this work, we address the questions of (1) whether the posterior distribution over matchings is approximable by a local algorithm, and (2) whether marginal statistics of this posterior have a well-defined limit as $n \to \infty$. We answer both questions affirmatively for partial matching, where a decay-of-correlations arises for large $n$. For exact matching, we show that the posterior is approximable locally only after a global sorting of the points, and that defining a large-$n$ limit of marginal statistics requires a careful indexing of points in the Poisson point process limit of the data, based on a notion of flow. We leave as an open question the extensions of such results to dimensions $d \geq 2$.