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

2026-03-09

AI Summary

This study investigates whether the Bayesian posterior distribution of a one-dimensional noisy random point set under an unknown matching π* can be approximated by local algorithms, and examines the limiting behavior of its marginal statistics as n → ∞. By integrating Bayesian inference, local algorithms, Poisson point process limits, and flow theory, the work establishes that under partial matching, the posterior exhibits decaying correlations, admits a local approximation, and possesses a well-defined infinite-volume limit. In contrast, for exact matching, a global ordering must first be imposed, after which a flow-based indexing mechanism enables the construction of the limiting object. These results lay a theoretical foundation for future extensions to higher dimensions and raise several open questions in the field.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion

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 $d \geq 2$.

Problem

• Bayesian inference
• planted matching
• posterior approximation
• infinite-volume limit
• decay of correlations

Innovation

• Bayesian inference
• planted matching
• local posterior approximation
• decay of correlations
• infinite-volume limit