Constructive Disintegration and Conditional Modes

2025-08-01

AI Summary

This paper addresses foundational issues in conditioning within Bayesian statistics, identifying a fundamental flaw in the conventional identification of conditional distributions with restrictions of density functions to observed subsets—particularly problematic on smooth manifolds, where disintegration of measures may lack mathematical justification. Method: The authors rigorously distinguish “restricted densities” from properly defined “disintegration densities”, construct explicit counterexamples demonstrating their substantial divergence, and integrate differential geometry with measure theory to develop a computable framework for disintegration densities on manifolds, including an explicit algorithm. Contribution/Results: They prove that the widely used “conditional mode” corresponds to the mode of the restricted measure—not the disintegration measure—leading to theoretical inconsistency in approximate Bayesian inference and Bayesian inverse problems. The work clarifies essential prerequisites for valid conditional modeling and establishes a rigorous mathematical foundation for Bayesian methods on high-dimensional and manifold-structured parameter spaces.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differentiable manifolds. Using our results, we provide a disturbingly simple example in which the restricted density and the disintegration density drastically disagree. Motivated by applications in approximate Bayesian inference and Bayesian inverse problems, we further study the modes of disintegrations. We show that the recently introduced notion of a "conditional mode" does not coincide in general with the modes of the conditional measure obtained through disintegration, but rather the modes of the restricted measure. We also discuss the implications of the discrepancy between the two measures in practice, advocating for the utility of both approaches depending on the modelling context.

Problem

• Formalizing conditioning in Bayesian statistics via measure disintegration
• Resolving discrepancies between restricted and disintegration densities
• Clarifying differences between conditional modes and disintegration modes

Innovation

• Formalizes disintegration of measures for conditioning
• Provides tools for constructing disintegrations on manifolds
• Compares conditional modes and disintegration modes