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This paper claims that a mixed additive–multiplicative noise inverse problem can be rewritten exactly with an auxiliary variable u = ξ f(θ), giving a factorized likelihood that makes Bayesian inference tractable without approximate likeliho

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2026-08-01 05:05 UTC pith:HF65IAAX

load-bearing objection The exact auxiliary-variable likelihood reformulation is correct and worth having; the adaptive MCMC kernel's missing stationarity proof is the soft spot that keeps me from fully trusting the UQ claims. the 2 major comments →

arxiv 2607.22330 v1 pith:HF65IAAX submitted 2026-07-24 stat.ME eess.SP

A Hierarchical Likelihood Model for Non-linear Inverse Problems under Additive and Multiplicative Noise

Nicolas Goeman , Pierre-Antoine Thouvenin , Pierre Chainais This is my paper
classification stat.ME eess.SP MSC 62F1565C0562P35
keywords hierarchical Bayesian modelmultiplicative noiseadditive noisenon-linear inverse problemscensored dataMCMCuncertainty quantificationauxiliary variable
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper attacks ill-posed inverse problems where observations are nonlinear functions of parameters, subject to both additive Gaussian noise and lognormal multiplicative noise, and censored below a detection threshold. It argues that when both noise sources matter, exact Bayesian inference is still possible by transforming the model, without approximation, into a two-stage hierarchy with an auxiliary variable u = ξ f(θ). The likelihood then factorizes into a censored Gaussian observation term and a lognormal transition term, replacing the intractable original likelihood with a tractable joint likelihood over data and latent variables. The paper provides an MCMC sampler and demonstrates on a synthetic astronomical imaging problem that this exact hierarchical model matches or improves on an interpolated approximate likelihood while requiring no calibration of extra hyperparameters, with better predictive performance as measured by expected log predictive density.

Core claim

The central claim is that model (1), despite its intractable likelihood, is statistically equivalent to the pair of equations (3), and that the auxiliary variable u decouples the nonlinear forward model from the mixture of noises. Concretely, the joint likelihood factorizes as π(Y, U | Θ) = ∏_{n,ℓ} π(y_{n,ℓ} | u_{n,ℓ}) π(u_{n,ℓ} | θ_n), where the first factor is a rectified Gaussian density accounting for censoring and the second is a lognormal density from the multiplicative noise. Inference can then target the joint posterior of (Θ, U), avoiding approximate likelihoods. The authors claim this formulation is exact, needs no calibration of transition hyperparameters in the likelihood, and ca

What carries the argument

The workhorse is the exact hierarchical reformulation (3): introducing the auxiliary variable u = ξ f(θ) splits the combined noise into a multiplicative stage (u given θ is lognormal) and an additive censored stage (y given u is Gaussian with rectification at the detection threshold). This turns the intractable likelihood into a product of two standard distributions. Around that identity, the sampler uses a checkerboard-style parallel update scheme, a local gradient-based proposal, a multiple-try global proposal for mode exploration, and—for non-censored pixels—a Gamma proposal for u whose parameters are matched to the conditional mode and curvature; censored pixels use ancestral sampling fr

Load-bearing premise

The load-bearing premise is that the adaptive MCMC kernel keeps the posterior as its stationary distribution, since the exact-likelihood reformulation alone does not guarantee that the sampler's samples are unbiased.

What would settle it

A concrete check: numerically integrate the original likelihood (1) over the multiplicative noise ξ for one pixel and band, and compare the result with the marginal likelihood of the hierarchical model (3) at the same parameter values; any difference beyond Monte Carlo error would falsify the exact-equivalence claim. A second, sampler-level check is to run the chain with adaptivity disabled and compare the resulting posterior samples; a shift in the empirical distribution would indicate that the adaptive proposal alters the stationary distribution.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the equivalence is exact, the interpolated likelihood and its calibration step become unnecessary; practitioners can use the hierarchical model directly across noise levels.
  • Uncertainty quantification, in the form of credible intervals, can be computed from posterior samples and mapped back to physical parameter space through the log-normalization.
  • The checkerboard parallel updates allow pixels to be processed in batches, so the added latent dimension per pixel carries only modest computational overhead.
  • The model's predictive performance, measured by expected log predictive density, is higher than that of additive, multiplicative, and interpolated likelihood approximations, with the gap widening as multiplicative noise increases.
  • The hierarchical construction generalizes to other noise sources by inserting additional latent levels, making it a template for future mixed-noise inverse problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exactness claim concerns the model, not the sampler: a natural check is whether the adaptive gradient-based proposal preserves the posterior as its stationary distribution, since the paper cites a reference rather than proving this.
  • The same auxiliary-variable trick should extend to multiplicative noise laws beyond lognormal, provided the conditional distribution of u given θ remains tractable—an extension that could be tested on other heavy-tailed noise models.
  • In censored regions, the proposal for u reduces to sampling from the lognormal prior π(u|θ), so uncertainty there is essentially prior-driven; an explicit study of how sensitive censored-region credible intervals are to that prior would sharpen the UQ guarantees.
  • A concrete external test: for a fixed pixel and band, numerically marginalize the original likelihood over ξ and compare with the marginal of the hierarchical model; a mismatch beyond Monte Carlo error would refute the claimed statistical equivalence.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a hierarchical reformulation of the non-linear inverse problem (1), in which the latent multiplicative-noise variable u_{n,ℓ}=ξ_{n,ℓ} f_ℓ(θ_n) is introduced so that the observation model becomes y=max(ω, u+ε). The authors show that the hierarchical likelihood factorizes as π(Y,U|Θ)=∏π(y|u)π(u|θ) and that marginalizing over U recovers the original model (1). They argue that this exact reformulation avoids the need for the calibrated approximate likelihood of Palud et al. [8]. An MCMC sampler is then proposed, combining a PMALA local kernel and an I-MTM global kernel, with a Gamma-based proposal for the auxiliary variables. The method is evaluated on synthetic astronomical data under three multiplicative-noise levels and censoring, and compared with interpolated, additive, and multiplicative likelihood models using MAE, credible-interval sizes, ELPD, and wall-clock time. The main claims are that the hierarchical model is statistically equivalent to the original model, that it gives more reliable uncertainty quantification, and that it is competitive or better than the approximate baselines without requiring likelihood-hyperparameter tuning.

Significance. The exact-likelihood reformulation is a genuine and useful contribution. If the inference machinery is correct, the paper provides a principled way to handle mixed additive/multiplicative noise and censoring in non-linear inverse problems without approximating the likelihood or tuning interpolation hyperparameters. The experimental protocol is careful: multiple noise levels, five random seeds, Monte Carlo ELPD evaluation against the true generative model, and a pairwise model-comparison analysis. The ELPD comparison is a particularly valuable part of the study and goes beyond point-estimate comparisons. However, the uncertainty-quantification claims rest on the MCMC sampler being able to sample the target posterior, and the paper does not currently establish this for the adaptive PMALA kernel used in Algorithm 1. The exact reformulation is sound, but the inference guarantee is the load-bearing gap.

major comments (2)
  1. [Algorithm 1, line 12; Section III; Section IV-D] The validity of the reported credible intervals and ELPD comparisons depends on the MCMC samples being drawn from π(Θ,U|Y). Algorithm 1 line 12 says only 'Update PMALA kernel parameters (see [8])', and Section IV-D fixes the RMSProp decay rate at α=0.5 for the whole run, so the adaptation does not diminish. Section III asserts convergence by citing Jones et al. [15], but that reference concerns non-adaptive conditional Metropolis-Hastings and does not cover adaptive kernels. Without a proof of diminishing adaptation or another validity condition for the adaptive PMALA kernel, the sampler is not guaranteed to have π(Θ,U|Y) as its stationary distribution. This is not an objection to the exact hierarchical reformulation, but it directly undermines the uncertainty-quantification claims, including the 95% credible intervals and the ELPD-based conclusions.
  2. [Abstract; Section V; Table I; Section V computing time] The abstract claims 'state-of-the-art results in terms of point-wise estimates and computing costs'. Table I does not support superiority in point estimates: for example, at σ_m=log(1.5), the hierarchical MAE for Θ·,0 is 11.55±0.23 versus 11.07±0.30 for the multiplicative model; for Θ·,1, 8.61±0.07 versus 8.44±0.07; at σ_m=log(2.0), the interpolated model is often better. The median computing time per iteration is 8.26 s for the hierarchical model versus 5.34 s for the interpolated model. The results support comparable point estimates with better predictive performance, but not superior point estimates or computing costs. The abstract and conclusion should be amended to avoid overclaiming.
minor comments (5)
  1. [Figure 4 caption] The caption says 'From top to bottom: ground truth, hierarchical, interpolated, additive, multiplicative', but Figure 4 displays credible-interval sizes; ground truth has no credible interval. Please correct the caption or the figure rows.
  2. [Equation (14)] The mixture proposal q_θ in (14) is written without normalization constants. The constants cancel in the MH ratio, but this should be stated explicitly, especially because the text says 'contributions of the modes are reweighted to have equal probabilities'.
  3. [Section V, discussion of Table II] The phrase 'exp(0.182)≈1.2 times more likely' is imprecise. The reported quantity is a mean ΔELPD, i.e., a mean log predictive density ratio, not a posterior model probability. Please rephrase as 'on average exp(0.182)≈1.2 times higher predictive density'.
  4. [Code availability] The text says 'Codes to reproduce the experiments are part of a forthcoming update of the BEETROOTS library'. Since the experiments are central to the paper, please provide a stable DOI or repository link at submission time rather than a 'forthcoming' statement.
  5. [Section II-B] The lognormal notation LogN(·) is used before its parameterization is fully spelled out. Please state explicitly that the parameters are the mean and variance of the logarithm, to avoid ambiguity with alternative lognormal conventions.

Circularity Check

0 steps flagged

No significant circularity: the hierarchical likelihood is an exact variable transformation, and the reported comparisons are evaluated against external synthetic ground truth.

full rationale

The paper's central modeling claim is the equivalence between the original observation model (1) and the hierarchical reformulation (3)-(5). This is a direct variable transformation: defining u_{n,ell}=xi_{n,ell} f_ell(theta_n) and writing y_{n,ell}=max(omega_{n,ell}, u_{n,ell}+epsilon_{n,ell}) reproduces (1) exactly when the lognormal and Gaussian noise terms are substituted, and marginalizing the auxiliary variable recovers the original likelihood. No parameter is fitted to make this equivalence hold; it is a mathematical identity. The acceptance steps of the MCMC sampler use standard Metropolis-Hastings ratios, and the Gamma proposal in Appendix A is fitted only to the proposal distribution, not to any reported prediction or uncertainty claim, and is corrected by the accept/reject step. The ELPD comparisons are computed on synthetic data generated from the true model (1), which is an external benchmark rather than a quantity derived from the model's own fitted outputs. The paper does cite prior work for the reparameterization and for the PMALA design, but the exact-likelihood reformulation does not depend on the correctness of those citations, so no load-bearing self-citation chain is present. The adaptive PMALA stationarity issue identified by the skeptic is a genuine missing proof about the sampler's convergence, but it is a correctness risk, not a circularity: it does not make any claimed derivation reduce to its own inputs. Overall, the derivation chain is self-contained and non-circular.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 1 invented entities

The central claim rests on standard modeling assumptions (lognormal/Gaussian noise, independence), plus manually set hyperparameters τ_d, δ and sampler tuning values. The auxiliary variable u is a mathematical device, not a new physical entity.

free parameters (3)
  • spatial regularization hyperparameter τ_d = 20 (all d)
    Manually tuned in Section IV.A 'to provide a good trade-off between smoothness and detail preservation'; directly shapes the posterior (8).
  • validity-constraint hyperparameter δ = 10^4
    Set in Section IV.A; controls the strength of the compact-support penalty in (7)-(8).
  • sampler tuning parameters (p, ε, η, α, M, M_NR, P) = p=0.5, ε=1e-2, η=1e-5, α=0.5, M=50, M_NR=5, P=10
    Hand-selected in Section IV.D and Appendix A; affect mixing and acceptance rather than the model, but are free choices not derived from data.
axioms (6)
  • domain assumption Observation model (1): y = max(ω, ξ g(φ)+ε) with ξ ~ LogN(-σ_m²/2, σ_m²), ε ~ N(0,σ_a²), independent; thresholds ω fixed/known.
    The exactness of the hierarchical likelihood is relative to this generative model (Section II.B).
  • domain assumption Conditional independence of observations across pixels n and bands ℓ given Θ.
    Used in the factored likelihood (4) and for checkerboard parallel Gibbs (Sections II.B, III.A).
  • domain assumption Prior (8) with squared-gradient smoothness and smooth compact-support penalty, hyperparameters τ, δ.
    Borrowed from [8]; determines the posterior being sampled (Section II.C).
  • domain assumption The costly physical forward operator g can be replaced by a differentiable surrogate f = g∘ν^{-1} (neural network) that is sufficiently accurate.
    Section II.A; PMALA proposals require gradients; experiments use surrogate f from [24].
  • ad hoc to paper The mode u* of the conditional (15) lies between the modes of the lognormal and Gaussian factors.
    Appendix A uses this to initialize Newton-Raphson; if violated, the Gamma fit degrades (efficiency, not MH validity).
  • standard math Metropolis-within-Gibbs with block updates converges under mild conditions; the adaptive PMALA kernel maintains the correct stationary distribution.
    Section III cites [15]; the adaptive step (Algorithm 1, line 12) is not proven in this paper and is the weakest link for uncertainty quantification.
invented entities (1)
  • Auxiliary latent variable u_{n,ℓ} = ξ_{n,ℓ} f_ℓ(θ_n) no independent evidence
    purpose: Decouples multiplicative from additive noise, making the likelihood factorize into lognormal and rectified-Gaussian terms (Eqs. 3-5).
    A data-augmentation device without physical meaning; marginalizing u recovers model (1). It does not add an unsupported physical entity.

pith-pipeline@v1.3.0-alltime-deepseek · 15579 in / 17317 out tokens · 161713 ms · 2026-08-01T05:05:19.564379+00:00 · methodology

0 comments
read the original abstract

Ill-posed inverse problems are encountered in numerous applications, possibly characterized by a highly non-linear forward model, both additive and multiplicative sources of noise, and censored data. In the absence of ground truth, uncertainty quantification is crucial to assess estimation reliability. This motivates the use of a Bayesian model and stochastic inference methods such as Markov Chain Monte Carlo algorithms. Problems combining all these challenges often lead to a complex and potentially multimodal posterior distribution, difficult to handle in practice. Approximate approaches have been proposed in the literature by either neglecting a source of noise or by using a tractable approximation of the likelihood function. These approaches either lead to an inaccurate model, or may require a complex calibration of the approximate likelihood. This paper proposes to tackle such problems with a general hierarchical Bayesian model and an efficient MCMC algorithm. The proposed formulation bypasses the need for calibrating the hyperparameters of an approximate model and is more versatile. The proposed method is assessed on a challenging scenario encountered in astronomy using synthetic data, in a variety of noise and censoring configurations. Comparisons are conducted against two baselines and a state-of-the-art method applicable in this context. The proposed approach is general and yields state-of-the-art results in terms of point-wise estimates and computing costs, with superior predictive performance. Results in the supplementary material further complete this comprehensive and rigorous model study. This work can serve as a guide for practitioners to select the best likelihood model according to their specific application.

Figures

Figures reproduced from arXiv: 2607.22330 by Nicolas Goeman, Pierre-Antoine Thouvenin, Pierre Chainais.

Figure 1
Figure 1. Figure 1: Impact of the amplitude of fl(θn) on the dominant noise regime in the pdf of the conditional distribution on un,ℓ. Top: π(un,ℓ | θ˜ (t+1) n ) dominates. Middle: both component distributions are significant. Bottom: π(yn,ℓ | un,ℓ) dominates. However, even for moderate values of σm, the conditional (15) can still be highly skewed when fℓ(θ˜ (t+1) n ) significantly differs from yn,ℓ. This situation may occur … view at source ↗
Figure 2
Figure 2. Figure 2: Examples of synthetic observations for (a) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction results (MMSE) for each parameter map [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Element wise CI size for each parameter map and [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: ∆ELPD maps for σm = log(1.5) and for each pair of likelihood models. The L lines are summed for visual purposes. First row: hierarchical vs interpola￾tion/multiplicative/additive (from left to right). Second row: interpolation vs multiplicative/additive, multiplicative vs addi￾tive (from left to right). Positive values (in green) indicate a better performance of the first model in the pair, while lower val… view at source ↗
Figure 6
Figure 6. Figure 6: Examples of fitted Gamma proposal distributions (red [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

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