Constructive Disintegration and Conditional Modes

Jon Cockayne; Marvin Pf\"ortner; Natha\"el Da Costa

REVIEW 3 major objections 4 minor 17 references

Constructive Disintegration and Conditional Modes

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that conditioning on a nonlinear observation is not the same as restricting the prior density to the observation fiber, and that the 'conditional modes' used in MAP approximation are modes of the wrong measure unless a…

desk verdict The mode-discrepancy result is real but only proved at regular values; the paper is mathematically careful, with framing that overstates scope. read the letter →

arxiv 2508.00617 v1 pith:NQCCYWPE submitted 2025-08-01 math.ST cs.LGmath.PRstat.MLstat.TH

Nathaël Da Costa , Marvin Pförtner , Jon Cockayne This is my paper

classification math.STcs.LGmath.PRstat.MLstat.TH MSC 60A1062F1558A10

keywords disintegrationofmeasuresconditionalmodesMAPestimationOnsager-MachlupfunctionalRiemannianmanifoldsBayesianinverseproblemsrestrictedco-areaformula

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Conditioning, formally a disintegration of measures, is hard to construct outside Gaussian-linear settings, so a folklore shortcut restricts the prior density to the observation fiber $h^{-1}(\{y\})$ and renormalizes. This paper claims that on a Riemannian manifold, and in particular in $\mathbb{R}^d$, the true disintegration density along a fiber is the prior density times a Jacobian-type factor $|\det(Dh(x)|_{\ker(Dh(x))^\perp})|^{-1}$, so restriction is generically wrong for nonlinear observations. The paper exhibits a two-dimensional Gaussian conditioned on ellipses where the restricted and disintegration densities disagree so strongly that their modes are nearly opposite. It further claims that recently introduced 'conditional modes' are modes of the restricted measure, not of the disintegration, and that true weak modes minimize $-\log(d\mu/d\lambda)+\log|\det(Dh|_{\ker(Dh)^\perp})|$ on $h^{-1}(\{y\})$. If this is right, existing MAP-style approximations on submanifolds are solving a different optimization problem, and the paper supplies the missing correction term.

What carries the argument

The load-bearing object is the local fiber-density factor $|\omega_{\ker(Dh(x))^\perp}(x)[\nabla h(x)]|^{-1}$, which in $\mathbb{R}^d$ is $|\det(Dh(x)|_{\ker(Dh(x))^\perp})|^{-1}$. It measures how densely the family of fibers $h^{-1}(\{y'\})$ sits near a point: where the gradient is small, fibers are spread further apart, and the disintegration must place more mass there to satisfy the law of total probability. The factor is exactly what converts the Riemannian restricted measure $\mu_{h^{-1}(\{y\})}$ into the true conditional. The mode argument then rides on exhaustive Onsager-Machlup functionals, which characterize weak modes as minimizers of a small-ball ratio limit and let the paper turn the corrected density into the constrained minimization problem of Theorem 1.2.

What would settle it

As a scope check, condition a standard Gaussian on $\mathbb{R}^2$ with $h(x)=x_1^2$ at the critical value $y=0$: the fiber is the line $\{x_1=0\}$ and a disintegration exists, but $Dh$ vanishes on it, so the paper's formula is not defined there. As a quantitative check of the mechanism, run the paper's ellipse example with $a=b$, where the Jacobian factor is constant: the restricted and disintegration densities should then agree up to normalization, and their modes should coincide, confirming that all discrepancy is carried by the factor the paper identifies.

Watch

Extended reading notes

Core claim

The central discovery is an explicit density formula for the disintegration of a measure along the fibers of a smooth observation map. Under $C^1$ regularity (Assumption 2.12), the paper constructs a version of the disintegration $\mu^h(\cdot|y)$ that is absolutely continuous with respect to the Riemannian volume measure $\omega_{h^{-1}(\{y\})}$ of the fiber and shows that $d\mu^h(\cdot|y)/d\omega_{h^{-1}(\{y\})}(x) \propto (d\mu/d\omega_X)(x)\,|\omega_{\ker(Dh(x))^\perp}(x)[\nabla h(x)]|^{-1}$. In $\mathbb{R}^d$ the factor is the absolute determinant of $Dh(x)$ restricted to the orthogonal complement of its kernel. The paper therefore states that renormalizing the restricted density is not conditioning. It also characterizes weak modes through exhaustive Onsager-Machlup functionals, concluding that the weak modes of the disintegration minimize the negative log density plus the log of the Jacobian factor on the fiber, while the 'conditional modes' introduced in prior work drop that factor and are modes of the restricted measure instead.

Load-bearing premise

The load-bearing premise is that $h$ is $C^1$ with surjective derivative at $\mu$-almost every point and that the conditioned value $y$ is a regular value of $h$; at critical values the paper's corrective factor is undefined and the fiber may not be a smooth submanifold, so the constructive formula does not cover cases where a disintegration still exists.

Editorial extensions

If this is right

For nonlinear or manifold-valued observations, renormalizing the prior density on the observation fiber does not yield the conditional distribution; the correct disintegration density includes the inverse-Jacobian factor.
Restricted-mode MAP estimators, including recently proposed conditional modes and probabilistic-ODE-solver MAP estimators, are generally not MAP estimators of the conditional distribution.
When the observation map is linear, the Jacobian factor is constant, so restricted modes and disintegration modes coincide and the classical density-conditioning formula is recovered.
For regular observations on a $C^2$ Riemannian manifold, weak modes of the disintegration are exactly the minimizers on the fiber of the Onsager-Machlup functional $I_{\mu_{h^{-1}(\{y\})}}(x)+\log|\omega_{\ker(Dh)^\perp}[\nabla h]|$.
A change of norm on $\mathbb{R}^d$ can change the modes of the disintegration even though the prior's modes are unchanged, because the fiber's small-ball volume depends on the fiber's orientation relative to the norm's unit ball.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One extension the authors leave implicit is a practical sampling correction: for a nonlinear observation map, reweighting fiber-restricted samples by $|\det(Dh|_{\ker(Dh)^\perp})|^{-1}$ should move them from the restricted measure to the disintegration.
The critical-value gap suggests that any infinite-dimensional or degenerate version of the result will need separate handling of singular fibers, since the gradient factor is not defined where the observation map's derivative drops rank.
If the ellipse discrepancy is representative, Laplace-type approximations built from restricted modes may report posterior location and curvature in the wrong region whenever the observation operator is nonlinear, a risk worth testing on moderate-dimensional inverse problems.
Comparing restricted and adjusted MAP estimates for probabilistic ODE solvers with a genuinely nonlinear residual would quantify how often the two prescriptions differ in practice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short take: the paper's real content is the mode analysis—showing that 'conditional modes' in Chen et al. and related work are modes of the restricted measure, not of the disintegration. That result is new and useful, and the mathematics behind it checks out. But the central claim is proved only at regular values of the observation map, and the abstract doesn't say that. The stress-test example is a fair hit: condition a Gaussian on h(x)=x1^2 at y=0, the fiber is a line, the disintegration exists, and the paper's constructive formula doesn't apply because Dh vanishes on the fiber. This doesn't sink the paper, but it needs to be stated as a scope condition.

What's genuinely new: Theorem 3.5 and Corollary 3.10 give an OM-functional characterization of weak modes of the disintegration on Riemannian manifolds, with the log-Jacobian correction term relative to the restricted mode. The norm-dependence result in Section 3.2 is a nice observation and not something I've seen stated cleanly. The example in Figure 1 is simple and the factors are computed correctly. The Euclidean density formula is a coarea formula statement and Diaconis et al. already have it; the authors acknowledge that, so the novelty claim should be trimmed.

Soft spots in order of size: (1) regular values only. Theorem 2.14 and the mode theorems all assume y is a regular value. Critical observations occur naturally—boundaries, squared coordinates at zero—and the paper's abstract and introduction make unqualified claims. This is a framing flaw, not a math flaw. (2) 'First general set of results' overstates the coarea-formula antecedents. (3) The Bishop von Mises remark is a bit unfair: on the unit circle the Jacobian factor is constant, so restriction and disintegration coincide; the example they use in Figure 1 is the right one, but calling Bishop's description an 'error' is too strong.

Bottom line: the paper is honest, careful, and the central mode-discrepancy result is a real and useful correction to recent practice. The main theorem is proved with standard tools and the lemmas are solid. It deserves a serious referee. The revision should qualify the scope to regular values, soften the novelty and the Bishop remark, and then it'll be a good contribution to the conditioning-on-submanifolds literature. I'd bring it to a reading group and I'd cite it (with the caveat).

Referee Report

3 major / 4 minor

Summary. The paper develops constructive tools for disintegrations of measures on Riemannian manifolds, culminating in an explicit formula for the density of a disintegration with respect to the Riemannian volume of the fiber (Theorem 2.14). It then uses Onsager-Machlup functional theory to characterize weak modes of the disintegration (Theorem 1.2, Corollary 3.10), and argues that the 'conditional modes' used in several recent works are actually modes of the restricted measure rather than of the disintegration. The paper also studies how the mode characterization changes in normed spaces and presents a numerical example contrasting restriction and disintegration.

Significance. If the main results are read with the qualifications discussed below, the paper makes a useful contribution: it gives an explicit, checkable formula for disintegrations on manifolds, clarifies a common conflation between restriction and conditioning, and connects the discrepancy to Onsager-Machlup functionals. The proof strategy via disintegration lemmas is elegant, and the elliptical example is a convincing illustration that the correction term can change the location of modes. The paper also ships a substantial amount of supporting geometry in the appendices. The novelty relative to coarea-formula-based constructions is real but incremental, and the main theorem statements need qualification before the advertised claims are fully supported.

major comments (3)

[Section 2.2, Theorems 1.1 and 2.14] The density formula is stated for all C^1 regular values y, but Lemma 2.9, which drives the proof, only yields the proportionality for h_*µ-almost every y. At a regular value y for which the prior density vanishes on the entire fiber, the right-hand side integrates to zero and cannot be normalized to a probability density; for example, take µ uniform on [0,1]^2 and h(x)=x_1 with y=2. The theorem should be restricted to h_*µ-almost every y, or to regular values y for which the fiber integral of (dµ/dω_X) times the inverse Jacobian factor is finite and positive. The same qualification propagates to Theorem 1.2 and Corollary 3.10.
[Abstract and Section 4] The discrepancy between restricted modes and modes of the disintegration is presented without qualification, but all mode characterizations (Theorems 1.2 and 3.5, Corollary 3.10, Proposition 3.16) are stated only for C^1 regular values of h. For a natural critical observation such as h(x)=x_1^2 at y=0, the corrective factor |ω_{ker(Dh)^⊥}[∇h]|^{-1} is undefined and the paper establishes no result. The abstract's unqualified wording therefore overstates the scope; please add a caveat and discuss whether the discrepancy persists for critical observations.
[Theorem 1.2 versus Corollary 3.10] Theorem 1.2 is stated under µ-almost-everywhere C^1 regularity of h, but its proof relies on Corollary 3.10, which requires Assumption 3.8 (C^2 regularity) and, through Proposition B.2, C^2 submanifolds. Either state Theorem 1.2 under the C^2 assumption or provide the small-ball volume asymptotics needed for C^1 fibers; as written, the introduction's central mode formula is not a direct consequence of the stated results in Section 3.

minor comments (4)

[Section 1] The claim of providing 'the first general set of results for constructing disintegrations' is too strong in view of the coarea-formula-based construction in Diaconis et al. [2013] and related work; please soften or clarify the novelty.
[Corollary 3.17] The symbol 'V_{d-n}(0)' should be 'V_{d-n}(x)', since the quantity depends on the tangent space of the fiber at x.
[Corollary 3.15] There is a stray period before the function '-log dµ/dω_X|_{h^{-1}({y})}' in the statement; it should be removed.
[Introduction, Theorem 1.2] The text says Theorem 1.2 is a special case of Theorem 3.5, but the explicit form involving -log(dµ/dλ) also uses Corollary 3.10 and Proposition 3.9; the attribution should be made more precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central disintegration formula and mode characterizations are derived from the defining axioms and external theory, not assumed.

full rationale

The paper's claimed derivation chain is self-contained: Theorem 2.14 derives the disintegration density from Definition 2.2, the Chang--Pollard disintegration theorem, the composition lemmas (Lemmas 2.6--2.10, Proposition 2.11), and standard Riemannian geometry (Appendix A); the Jacobian correction is computed so that the law of total probability holds, not imposed as an input. The mode results (Theorem 1.2, Corollary 3.10) apply Theorem 2.14 and import the exhaustive Onsager--Machlup characterization from Ayanbayev et al. (2022), an external result, and the proof that Chen et al.'s 'conditional mode' is a restricted mode is established via Propositions 3.12, 3.14, and Corollary 3.15 from the ball-ratio definitions. No fitted parameter is renamed as a prediction, and no load-bearing premise is justified only by a self-citation; Cockayne et al. (2019) and Cinquin et al. (2024) appear only as motivating examples. The restriction of the main theorem to C1 regular values is a scope limitation, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central derivation relies on standard measure-theoretic disintegration theory, the implicit function theorem, the coarea formula in local coordinates, and small-ball volume asymptotics for Riemannian and normed spaces. No free parameters are fitted. No new entities are postulated. The main scope-limiting assumption is the C^k regularity of the observation map (non-critical values only).

assumptions (7)

standard math Disintegration theorem (existence and h⋆µ-a.e. uniqueness), Chang and Pollard 1997, Theorem 1.
Invoked in Theorem 2.4 as the foundation for all disintegrations.
standard math Implicit function / preimage theorem (Theorem A.9): for regular points the fiber is a submanifold with local chart φ with π_V ∘ φ = h.
Used in the proof of Theorem 2.14 to reduce the fiber to a product chart.
standard math Pushforward of the Riemannian volume measure through a chart equals |ω_U[∇φ∘φ^{-1}]|^{-1} λ (Eq. A.4).
Core change-of-variables identity in the proof of Theorem 2.14.
standard math Small-ball volume expansions: ω_X(B_r(x)) = V_d r^d + o(r^d), ω_{fiber}(B_r^X(x)) = V_{d-n} r^{d-n} + o(r^{d-n}) (Props B.1, B.2), and the normed-space analogue (Prop B.3).
Underpins all OM-functional and mode characterizations in Section 3; requires C^2 regularity.
standard math Weak modes equal minimizers of exhaustive OM functionals (Ayanbayev et al. 2022, Prop 4.1).
Used in Theorems 3.4, 3.5 and Corollaries 3.10, 3.15.
domain assumption Assumption 2.12: h is C^k regular at µ-a.e. points and µ is absolutely continuous w.r.t. a volume measure.
Scope-limiting assumption: excludes conditioning at critical values of h where fibers are not smooth submanifolds.
standard math Lemma C.2: reweighting by a continuous positive density updates an OM functional by -log f.
Used to pass from restricted measures to disintegrations; attributed to Ayanbayev et al. 2022 Lemma B.8.

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Cite this review

Pith. "Pith review of Constructive Disintegration and Conditional Modes." pith.science (2026). https://pith.science/paper/NQCCYWPE

@misc{pith2026250800617,
  author       = {Pith},
  title        = {Pith review of: Constructive Disintegration and Conditional Modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQCCYWPE}},
  note         = {Machine review of arXiv:2508.00617}
}

read the original 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.

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Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [6]

doi:10.1137/17M1139357

ISSN 0036-1445, 1095-7200. doi:10.1137/17M1139357. URL https://epubs.siam.org/doi/10.1137/ 17M1139357. M. Dashti, K. J. H. Law, A. M. Stuart, and J. Voss. MAP estimators and their con- sistency in Bayesian nonparametric inverse problems.Inverse Problems, 29(9),

work page doi:10.1137/17m1139357

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URL https://projecteuclid

doi:10.1214/12-IMSCOLL1006. URL https://projecteuclid. org/ebooks/institute-of-mathematical-statistics-collections/ Advances-in-Modern-Statistical-Theory-and-Applications--A-Festschrift/ chapter/Sampling-from-a-Manifold/10.1214/12-IMSCOLL1006. Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine. Riemannian Geometry. Universitext. Springer, Berlin, H...

doi:10.1214/12-imscoll1006

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doi:10.1007/s11222-019-09900-1

ISSN 1573-1375. doi:10.1007/s11222-019-09900-1. URL http://dx.doi.org/10.1007/ s11222-019-09900-1. Filip Tronarp, Simo Särkkä, and Philipp Hennig. Bayesian ODE solvers: the maximum a posteriori estimate. Statistics and Computing, 31(3), May

doi:10.1007/s11222-019-09900-1

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doi:10.1007/s11222-021-09993-7

ISSN 0960-3174, 1573-1375. doi:10.1007/s11222-021-09993-7. URL https://link.springer.com/10. 1007/s11222-021-09993-7. Publisher: Springer Science and Business Media LLC. Jonas Gregor Wiese, Lisa Wimmer, Theodore Papamarkou, Bernd Bischl, Stephan Günnemann, and David Rügamer. Towards efficient mcmc sampling in Bayesian neural networks by exploiting symmetr...

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ISBN 978-3-031-43412-9

Springer Nature Switzerland. ISBN 978-3-031-43412-9. 30 Appendix A. Differential Geometry Background Here we include the necessary background in differential and Riemannian geometry. For a more complete exposition to this background, we refer to Gallot et al. [2004]. Let X be a d-dimensional Ck manifold, with k ≥

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Similarly the inclusionexp˜X x (BRd r−o(r)(0) ∩ Tx ˜X) ⊂ BRd r (x) ∩ ˜X yields the reverse inequality ω˜X(BRd r (x)) ≥ V ˜d(x)r ˜d + o(r ˜d) as r ↓ 0, which concludes the proof. C. Modes Lemmas Lemma C.1(Ayanbayev et al., 2022, Lemma B.1(a)). An OM functionalIµ : Eµ → R is exhaustive if and only if there isx2 ∈ Eµ such that lim r↓0 µ(Br (x1)) µ(Br (x2)) =...

work page 2022

[7] [1997]

doi:10.1111/1467-9574.00056

ISSN 0039-0402, 1467-9574. doi:10.1111/1467-9574.00056. URL https://onlinelibrary.wiley.com/doi/10.1111/1467-9574.00056. Yifan Chen, Bamdad Hosseini, Houman Owhadi, and Andrew M. Stuart. Solving and learning nonlinear PDEs with Gaussian processes.Journal of Computational Physics, 447:110668, December

arXiv

[8] [1999]

doi:10.1002/9780470316979

ISBN 9780470316979. doi:10.1002/9780470316979. URLhttp://dx.doi.org/10.1002/ 9780470316979. Renata Possobon and Christian S. Rodrigues. Geometric properties of disintegra- tion of measures, February

doi:10.1002/9780470316979