Recognition: unknown
Response to: "A note on conditional densities, Bayes' rule, and recent criticisms of Bayesian inference" by Yan et al., 2026
Pith reviewed 2026-05-07 07:32 UTC · model grok-4.3
The pith
Conditional expectations do not fix physical inconsistencies in Bayesian methods for physical problems.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Yan et al. address only statistical consistency of Bayesian solutions under variable changes, which is already resolved by conditional expectations, but do not address physical consistency under changes in derivation mathematics. The theory of conditional expectations does not resolve physical inconsistency, Yan et al. make mathematical errors, and their critique is unfounded in several cases. Therefore the conclusions of the original preprint stand.
What carries the argument
The separation between physical consistency (solution invariance when the mathematics of derivation changes) and statistical consistency (solution invariance when variables change), which shows that the critique targets a different issue than the one raised in the original work.
If this is right
- Commonly used Bayesian posterior computations remain physically inconsistent for physical world applications.
- Hierarchical Bayesian methods with acausal structures continue to be invalid in this context.
- Practitioners in geophysics and physics must account for this inconsistency when choosing inference methods.
- The original preprint's examples of inconsistency are not refuted by the use of conditional expectations.
Where Pith is reading between the lines
- If physical consistency is required, Bayesian methods may need to be replaced or fundamentally modified for scientific modeling.
- This distinction might apply to other probabilistic methods used in physics beyond Bayes.
- Testing specific physical problems with different derivation paths could reveal the inconsistency explicitly.
Load-bearing premise
The distinction between physical consistency under changed derivation mathematics and statistical consistency under changed variables is both meaningful and decisive for whether Bayesian methods are valid in the physical world.
What would settle it
Demonstrate a physical problem where using conditional expectations to compute a Bayesian posterior yields the same result under different variable choices but produces different results when the same posterior is derived using two different mathematical approaches.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This manuscript is a direct response to Yan et al. (arXiv:2603.27038v1), who claim that the theory of conditional expectations eliminates inconsistencies in classical conditional probability densities identified in the authors' prior preprint (Mosegaard and Curtis, arXiv:2411.13570v2). The authors argue that Yan et al. address only statistical consistency under change of variables (already known to be resolved by conditional expectations) while failing to engage physical consistency under change of the underlying mathematics used to derive a solution. They assert that Yan et al. commit mathematical errors, that their critique is in places meaningless, and that the original preprint's conclusions on the invalidity of certain Bayesian methods for physical problems therefore stand.
Significance. If the claimed distinction between physical and statistical consistency can be made rigorous with explicit derivations or counter-examples, and if the identified mathematical errors in Yan et al. are substantiated, the work would reinforce concerns about acausality and inconsistency in hierarchical Bayesian methods when applied to physical systems. The paper correctly notes that measure-theoretic tools resolve variable-change issues but does not yet demonstrate that they leave the original physical-inconsistency examples intact; establishing this would be a substantive contribution to the ongoing debate on foundations of Bayesian inference in the physical sciences.
major comments (2)
- [Abstract / main argument] The central assertion that 'the theory of conditional expectations does not resolve physical inconsistency' (abstract) is load-bearing for the claim that Yan et al.'s note fails to address the preprint's tenet, yet the manuscript provides no explicit re-derivation of the original physical-inconsistency example under the conditional-expectation framework, nor a concrete numerical or measure-theoretic counter-example showing differing physical predictions. This leaves the distinction between the two consistency types as an assertion rather than a demonstrated separation.
- [Abstract / response to Yan et al.] The manuscript states that Yan et al. 'make mathematical errors' and that 'in some cases we show that their critique is meaningless,' but does not identify the specific equations, definitions, or steps in Yan et al. that are erroneous. Without these citations or corrections, the claim that the critique is unfounded cannot be evaluated independently of the prior preprint.
minor comments (2)
- [Abstract] The abstract refers to 'our preprint' and 'Yan et al., 2026' without a full bibliographic entry or arXiv identifier for the latter in the opening paragraph; this should be supplied for clarity.
- [Introduction] The distinction between 'physical consistency' and 'statistical consistency' is introduced without a formal definition or reference to prior literature on the topic; a brief definitional paragraph would aid readers unfamiliar with the authors' earlier work.
Simulated Author's Rebuttal
We thank the referee for their careful reading and constructive comments on our response note. We address each major comment below and indicate where revisions will be incorporated to strengthen the manuscript.
read point-by-point responses
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Referee: [Abstract / main argument] The central assertion that 'the theory of conditional expectations does not resolve physical inconsistency' (abstract) is load-bearing for the claim that Yan et al.'s note fails to address the preprint's tenet, yet the manuscript provides no explicit re-derivation of the original physical-inconsistency example under the conditional-expectation framework, nor a concrete numerical or measure-theoretic counter-example showing differing physical predictions. This leaves the distinction between the two consistency types as an assertion rather than a demonstrated separation.
Authors: We agree that an explicit re-derivation or counter-example in this note would make the distinction between statistical and physical consistency more rigorous and self-contained. The original physical-inconsistency examples appear in Mosegaard and Curtis (arXiv:2411.13570v2), where classical conditional densities produce physically inconsistent posteriors under change of variables. Yan et al. resolve only the statistical consistency issue via conditional expectations. In the revised manuscript we will add a concise section that outlines the original example in measure-theoretic terms, shows that conditional expectations preserve statistical consistency under reparameterization, and demonstrates that the physical predictions (e.g., support of the posterior and implied causal structure) nevertheless remain inconsistent when the underlying mathematical representation is altered. This addition will include a brief numerical illustration to separate the two notions of consistency. revision: yes
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Referee: [Abstract / response to Yan et al.] The manuscript states that Yan et al. 'make mathematical errors' and that 'in some cases we show that their critique is meaningless,' but does not identify the specific equations, definitions, or steps in Yan et al. that are erroneous. Without these citations or corrections, the claim that the critique is unfounded cannot be evaluated independently of the prior preprint.
Authors: We accept that the current text would benefit from explicit pointers to the claimed errors. In the revised version we will cite the precise equations and definitions in Yan et al. (arXiv:2603.27038v1) that we regard as erroneous or incomplete when applied to physical problems—for example, their treatment of the joint density under conditional expectation and the resulting marginals—and explain why those steps fail to preserve physical consistency. We will also clarify the passages in which their critique is meaningless by contrasting the statistical resolution they provide with the physical requirements of the original problem (change of the underlying mathematics rather than change of variables). revision: yes
Circularity Check
Central claim that prior conclusions stand reduces to self-cited preprint via asserted distinction between consistency types
specific steps
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self citation load bearing
[Abstract]
"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 ... The conclusions of our preprint therefore stand."
The paper's argument that conditional expectations leave the claimed physical inconsistency intact rests on the existence and nature of that inconsistency as demonstrated in the authors' own prior preprint (Mosegaard & Curtis 2024, arXiv:2411.13570v2). No new derivation or concrete example is exhibited here showing that the inconsistency survives re-derivation via conditional expectations; the response instead re-asserts the prior distinction and concludes the original claims are untouched.
full rationale
The response asserts that Yan et al. address only statistical consistency (resolved by conditional expectations) while the original work addressed physical consistency under change of derivation mathematics, so the critique fails to engage the tenet and prior conclusions stand. This distinction and the demonstration of persisting physical inconsistency are taken directly from the self-cited 2024 preprint without an independent re-derivation, explicit counter-example, or measure-theoretic argument supplied in the present text. The load-bearing step is therefore the self-citation chain rather than a self-contained derivation.
Axiom & Free-Parameter Ledger
Reference graph
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discussion (0)
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