Response to: "A note on conditional densities, Bayes' rule, and recent criticisms of Bayesian inference" by Yan et al., 2026

2026-04-30

AI Summary

This study clarifies the controversy surrounding physical inconsistency in Bayesian inference and addresses misconceptions raised by Yan et al. regarding prior work. Specifically, it refutes their claim that conditional expectation resolves Bayesian inconsistency by demonstrating that they conflate statistical consistency with physical consistency. Through rigorous analysis within a measure-theoretic framework—combining conditional expectation theory and transformation properties under variable reparameterization—the paper proves that mainstream Bayesian methods remain physically inconsistent under variable transformations. It further shows that conditional expectation cannot resolve this fundamental issue, corrects mathematical errors in Yan et al.’s argument, and reaffirms the original conclusion: Bayesian posteriors lack invariance under reparameterization in physical modeling contexts.

Abstract

In a recent preprint (Mosegaard and Curtis, 2024, arXiv:2411.13570v2) we analyzed the consequences of ignoring the well-known inconsistency of classical conditional probability densities. We explained how this inconsistency, together with acausality in hierarchical methods, invalidate a variety of commonly applied Bayesian methods when applied to problems in the physical world. Yan et al., 2026, (arXiv:2603.27038v1) published a note, in which they claim, contrary to our preprint, that there are no inconsistencies if one uses the method of conditional expectations to derive probabilities. Furthermore, they believe that there are mathematical errors in our exposition and in our use of the Bayesian framework. This note is a response to the claims made by Yan et al. Yan et al. do not discriminate between physical and statistical consistency. Their note addresses statistical consistency of a solution under a change of variables; this is already known to be resolved by using the theory of conditional expectations. By contrast, our preprint concerns the physical consistency of any solution under a change of mathematics used to derive that solution. It demonstrates that widely used methods to compute Bayesian posterior solutions are physically inconsistent under a change of variables. Their note does not, therefore, address the tenet of our preprint. We show herein that the theory of conditional expectations does not resolve physical inconsistency, and that Yan et al. make mathematical errors. We conclude that their claims are unfounded, and in some cases we show that their critique is meaningless. The conclusions of our preprint therefore stand.

Problem

• Bayesian inference
• physical consistency
• change of variables
• conditional expectations
• posterior computation

Innovation

• physical consistency
• conditional expectations
• Bayesian inference
• change of variables
• conditional probability densities