REVIEW 2 major objections 3 minor 15 references
Random Inverse Problems with Structural and Probabilistic Ambiguities
T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper claims that in random inverse problems, resolving probabilistic ambiguity depends on the observation scenario: when each observation carries its own independent draw of the random forward-model parameters, Bayesian inversion colla
desk verdict Useful taxonomy and clean derivations, but the headline claim about scenario (i) resolving probabilistic ambiguity is not general and needs an identifiability condition. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the random-equation likelihood $L(0\mid x) = f_{M(x;A)+B-y}(0)$: the density of the residual random variable evaluated at zero, marginalized over the parameter density $f_A$ by the law of total probability. For scenario (i) the marginalization sits inside a product over independent observations, $L^{(i)}(0\mid x) = \prod_{l=1}^{L} \int f_{B_l}(\hat y_l - M(x;s)) f_A(s)\,ds$; for scenario (ii) it is a single integral over the shared $A$; for scenario (iii) it becomes $\int f_{\hat y}(M(x;s)) f_A(s)\,ds$. The placement of this marginalization is what makes the probabilistic ambiguity resolvable or not. The forward model is arbitrary, nonlinear, and only piecewise continuo
What would settle it
Simulate the 1D model $y = A_1 x^2 + B$ under scenario (i) with a tri-modal $f_{A_1}$ and a large number of observations, but draw the $A_1$ values from a correlated process with the same marginal modes, for example a Markov chain. If the posterior still shows only the two structural peaks at $\pm x$, the independence assumption is not load-bearing; if extra parameter-induced peaks or broadening appear, the resolution claim fails. A companion check repeats the experiment with a misspecified $f_A$, shifted or narrowed components, and looks for posterior peak shifts or mode splitting.
Extended reading notes
Core claim
On the paper's own terms, the discovery is a scenario-dependent separation of two ambiguities. The forward model $y = M(x;A) + B$ is non-injective, so one observed $y$ admits many $x$; this is structural ambiguity. The parameter vector $A$ is assigned a mixture-model density with separated components, so the output itself carries a combinatorial multimodality; this is probabilistic ambiguity. The paper derives posterior densities by marginalizing the likelihood of the residual $M(x;A) + B - y$ at zero over $A$, using Monte Carlo integration, and applies the result to quadratic models. Numerically, scenario (i), where each observation has its own draw $A_l$, yields a posterior whose peaks are
Load-bearing premise
The scenario-(i) resolution stands on the assumption that the forward-model parameter draws $A_l$ are independent and identically distributed across observations with a known density $f_A$; if the draws are correlated or the density is misspecified, the marginalization step that averages over $A_l$ no longer collapses the probabilistic ambiguity.
Editorial extensions
If this is right
- In scenario (i), practitioners can quote a posterior with only structural ambiguity: mixture-model uncertainty in the forward parameters is fully averaged out, so multimodal parameter densities do not create extra input modes.
- In scenario (ii), a point estimate of the input is misleading: the posterior is a combinatorial mixture of structural and parameter modes, so uncertainty must be reported as a full multimodal distribution.
- Observing a full output density, scenario (iii), gives the same qualitative structural reconstruction as scenario (i) but with broader, less distinct peaks, so density observations carry less resolving power than independent samples.
- The Monte Carlo likelihood estimators work for arbitrary input and output dimensions and for piecewise continuous forward models, so the method is not tied to quadratic examples.
- The mixture components of $A$ can be read as simultaneously encoding forward-model ambiguity and uncertainty, giving a unified language for both sources of ill-posedness.
Reading between the lines
- A testable extension suggested by the scenario-(i) result: when the parameter $A$ enters the forward model linearly or additively, the marginalization over $f_A$ may be replaceable by evaluating $M$ at the conditional mean of $A$, which would cut Monte Carlo cost dramatically; the quadratic model $y = A_1 x^2 + B$ is the simplest test case.
- The scenario-(ii) backpropagation implies an identification warning that the paper does not state: with a single hidden parameter set, separate estimation of $x$ and $A$ is needed, for example by joint hierarchical inference, because the posterior cannot be decomposed into an input part and a parameter part.
- Scenario (iii) contrasts with stochastic inverse problems that push an input density forward; one could compare the two answers on the same quadratic models and measure how much the posterior differs from the push-forward density.
- The credibility gap between scenarios (i) and (iii) could be quantified by computing posterior entropy or peak sharpness on the same forward model under both observation types; the paper's plots suggest but do not measure this gap.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper formulates a class of inverse problems in which the forward map depends nonlinearly on a random parameter vector A and additive noise B. For three observation scenarios — (i) independent per-observation random parameters, (ii) a single global random parameter, (iii) an observed output density — the author derives marginal likelihoods by integrating out A and gives Monte Carlo algorithms for grid-based posterior computation. The central claim is a taxonomy of ambiguity: scenario (i) can resolve probabilistic ambiguities, leaving only structural ambiguities due to non-injectivity of the forward map; scenario (ii) backpropagates multi-modality of the parameter into the posterior; and scenario (iii) resolves probabilistic ambiguity with lower credibility. The derivations are applied to 1D and 2D quadratic models.
Significance. If the central claim held as stated, the three-scenario taxonomy would be a useful organizing perspective for inverse problems with random forward models, and the random-equation formalism provides a clean latent-variable marginalization. The likelihood derivations (Eqs. (5), (10), (13)) are correct under the stated independence assumptions, and the algorithms are directly implementable. However, the distinction between structural and probabilistic ambiguity is not formally sustained, and the numerical evidence is too narrow to validate the classification. The paper is best viewed as a proposal that requires an identifiability condition or a revised definition of structural ambiguity.
major comments (2)
- [Section 2.1, Eq. (5), Section 4] The definition of structural ambiguity as pointwise non-injectivity of M(·;s) is inconsistent with the scenario-(i) posterior. In Eq. (5), the posterior depends on x only through the marginal densities g_x(y)=∫ f_B(y−M(x;s))f_A(s) ds. Two inputs x and x' with g_x=g_{x'} are indistinguishable even if M(x;s)≠M(x';s) for every s in the support of f_A. Example: M(x;A)=A x, f_A=0.5δ_{−1}+0.5δ_1, B∼N(0,σ²). For s=±1, M(·;s) is injective, yet g_x=g_{−x}. Algorithm 1(i) would produce a two-peaked posterior at ±x_true, a residual ambiguity that is not structural under the paper's definition. The quadratic examples in Section 3 do not expose this because the pointwise preimage set of x→x² coincides with the f_A-invariance. To support the claim 'leaving only structural ambiguities,' the paper must either add an identifiability condition ensuring that g_x determines x up to the pointwise preimage se
- [Section 3, Figures 1–3] The numerical demonstrations use only quadratic forward models whose pointwise symmetries (x→−x, radial symmetry) coincide with the symmetry of the marginal likelihood. Consequently, the experiments are consistent with the derivations but cannot distinguish the proposed 'resolution of probabilistic ambiguity' from a coincidence of symmetries. A non-quadratic example, such as M(x;A)=A x with symmetric f_A, would directly expose the issue raised above. Additionally, the paper does not define a quantitative criterion for 'resolved.' A formal statement (e.g., convergence of the posterior to the structural equivalence class as L→∞, or a measure of posterior concentration) would make the central claim testable rather than qualitative.
minor comments (3)
- [Section 2.1] Typesetting artifacts: superscripts are missing, e.g., 'y∈R R' should be y∈R^R and 'x∈R n' should be x∈R^n. Also the affiliation contains 'M¨ unchen' with a misplaced combining diaeresis.
- [Section 2.3] Latin hypercube sampling is mentioned for drawing from f_A, but LHS is usually defined on the unit cube and needs a transformation for arbitrary densities. Please specify the implementation.
- [Figure 2, third row] The construction of the observed density f_ŷ from samples is not fully described; please state the estimation method (e.g., kernel density, histogram) and any parameters.
Circularity Check
No circularity: posterior derivations are self-contained Bayesian marginalizations; self-citations are contextual and not load-bearing.
full rationale
The derivation chain is self-contained. The likelihoods for the three scenarios are obtained by elementary marginalization over the random parameter A: Eq (5) for i.i.d. per-observation A_l, Eq (10) for a single global A, and Eq (13) for an observed output density. These are standard applications of the law of total probability and are fully written out in the paper; the citation to [6] for the 'random equation' formalism is contextual, and [7]-[9] are only mentioned as related applications. No parameter is estimated from the data and then relabeled as a prediction: f_A and f_B are specified a priori, and the output samples/densities are generated from the same forward model in the numerical sections. The scenario-dependent ambiguity behavior in Section 4 is a mathematical consequence of whether A is per-observation or shared, not an artifact of a fitted quantity. The only identified concern—that scenario (i) resolution relies on A_l being i.i.d. with known f_A—is an explicit modeling assumption, not a circular step. Therefore no circularity.
Assumptions & free parameters
free parameters (1)
- Prior region Ω (uniform on grid) =
hand-chosen, not specified numerically
assumptions (4)
- domain assumption The forward model M is known and piecewise continuous, with uncertainty only through the random vector A.
- domain assumption In scenario (i), A_l are i.i.d. across observations; in scenario (ii), A is shared but B_l are independent; in scenario (iii), the observed density f_yhat is known.
- standard math The likelihood can be defined as the density of the difference random variable M(x;A)+B-y evaluated at 0, and the law of total probability applies.
- ad hoc to paper Equation (13) treats the observed output density f_yhat as a valid likelihood factor in scenario (iii).
Cite this review
Pith. "Pith review of Random Inverse Problems with Structural and Probabilistic Ambiguities." pith.science (2026). https://pith.science/paper/5CTSRUBS
@misc{pith2026260801439,
author = {Pith},
title = {Pith review of: Random Inverse Problems with Structural and Probabilistic Ambiguities},
year = {2026},
howpublished = {\url{https://pith.science/paper/5CTSRUBS}},
note = {Machine review of arXiv:2608.01439}
}
read the original abstract
In this concise paper, we investigate a computational class of random inverse problems that incorporates model uncertainties through random variable parameters nonlinearly in the forward model as well as additive observational uncertainty. Random inverse problems with nonlinear parameter dependencies may arise in engineering, geophysics, image processing or uncertainty quantification. We introduce a new perspective on structural ambiguities due to the non-injectivity of the forward model together with probabilistic ambiguities by assigning mixture model densities with separate components to the parameters, which leads to a possibly complex forward model, observation model and posterior. As a result, the mixture-model parameters in the forward model can be interpreted as simultaneously describing aspects of both the nonlinear ambiguity and the uncertainty of the inverse problem. The underlying solution algorithm is presented based on Bayesian inversion for three observation scenarios leading to posterior densities for the input for given output samples or an observed output density. By applying the derived algorithms to 1D and 2D quadratic models, we numerically demonstrate in which scenarios the proposed algorithm can resolve probabilistic ambiguities in the solution of the random inverse problem. It is demonstrated that making the residual structural ambiguities visible in the posterior and showing the interplay with probabilistic ambiguities is a relevant perspective, including cases with a finite and an infinite number of solutions.
Figures
Reference graph
Works this paper leans on
-
[1]
D. Calvetti, E. Somersalo, Inverse problems: From regularization to Bayesian inference, WIREs Comput. Stat. 10 (2018) e1427. https://doi.org/10.1002/wics.1427
-
[2]
Stuart, Inverse problems: A Bayesian perspective, Acta Numer
A.M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer. 19 (2010) 451–559. https://doi.org/10.1017/S0962492910000061
-
[3]
M. Dashti, A.M. Stuart, The Bayesian Approach to Inverse Problems, in: R. Ghanem, D. Higdon, H. Owhadi (Eds.), Handbook of Uncertainty Quantification, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-12385-1 7
-
[4]
K. Cranmer, J. Brehmer, G. Louppe, The frontier of simulation-based inference, Proc. Natl. Acad. Sci. USA 117 (2020) 30055–30062. https://doi.org/10.1073/pnas.1912789117
-
[5]
Sun, Local estimators and Bayesian inverse problems with non-unique solutions, Appl
J. Sun, Local estimators and Bayesian inverse problems with non-unique solutions, Appl. Math. Lett. 132 (2022) 108149. https://doi.org/10.1016/j.aml.2022.108149
-
[6]
W. Hoegele, Combinatorial potential of random equations with mixture models: Modeling and simula- tion, Math. Comput. Simulation 239 (2026) 696–715. https://doi.org/10.1016/j.matcom.2025.07.033
-
[7]
W. Hoegele, A full-density approach to simulating random iteration equations with applications, Com- mun. Nonlinear Sci. Numer. Simul. 163 (2026) 110508. https://doi.org/10.1016/j.cnsns.2026.110508
-
[8]
W. Hoegele, S. Brockhaus, Errors-in-variables model fitting for partially unpaired data utilizing mix- ture models, Statistics 59 (2025) 247–275. https://doi.org/10.1080/02331888.2024.2432564. 11
arXiv 2025
Show all 15 references
-
[9]
Hoegele, A Stochastic-Geometrical Framework for Object Pose Estimation Based on Mix- ture Models Avoiding the Correspondence Problem, J
W. Hoegele, A Stochastic-Geometrical Framework for Object Pose Estimation Based on Mix- ture Models Avoiding the Correspondence Problem, J. Math. Imaging Vision 66 (2024) 822–838. https://doi.org/10.1007/s10851-024-01200-2
2024 doi
-
[10]
Gelman, J
A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, D. B. Rubin, Bayesian Data Analysis, Third Edition, Chapman and Hall/CRC (2014). https://doi.org/10.1201/b16018
2014 doi
- [11]
- [12]
- [13]
-
[14]
Marzouk, H.N
Y.M. Marzouk, H.N. Najm, L.A. Rahn, Stochastic spectral methods for efficient Bayesian solution of inverse problems, J. Comput. Phys. 224 (2007) 560–586. https://doi.org/10.1016/j.jcp.2006.10.010
2007 doi
-
[15]
Fast Bayesian inversion for high dimensional inverse problems
Kugler, B., Forbes, F., Dout´ e, S. Fast Bayesian inversion for high dimensional inverse problems. Stat Comput 32, 31 (2022). https://doi.org/10.1007/s11222-021-10019-5 12
2022 doi
Reviewed August 6, 2026 · model on record in the stance chip above.
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