Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals

D. Delgado-Gomez; E. Curbelo; F. Llorente; L. Martino

REVIEW 4 major objections 4 minor 70 references

Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read An adaptive importance sampler, ATAIS, jointly infers a nonlinear model's parameters and its noise covariance, then recycles the same particles to approximate the full posterior without extra model evaluations.

desk verdict Useful first-stage sampler and a nice recycling trick, but the claimed joint posterior over theta and Sigma is not validated as written - worth refereeing, with diagnostics or a softened claim required. read the letter →

arxiv 2501.01148 v1 pith:RTURVDWB submitted 2025-01-02 stat.CO cs.CEstat.ML

E. Curbelo , L. Martino , F. Llorente , D. Delgado-Gomez This is my paper · ORCID

classification stat.COcs.CEstat.ML MSC 62F1562H1265C05

keywords Bayesianinversionadaptiveimportancesamplingcovariancematrixuncertaintyanalysismultioutputsignalstemperingsequenceofposteriors

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

This paper tries to make joint Bayesian inference over two coupled unknowns tractable: the parameters theta of a nonlinear multi-output model and the covariance matrix Sigma of the observation noise. Its proposal, complete ATAIS, splits the joint space into two blocks, runs adaptive importance sampling over theta while updating Sigma with an analytic maximum-likelihood formula, and then reuses the same theta particles to build an approximation of the full posterior over {theta, Sigma} without any additional evaluation of the expensive model. If it works, the payoff is credible intervals on covariance components, model selection via marginal likelihood, and uncertainty analysis in problems where direct sampling on the joint space is impractical. The paper also introduces a simpler competitor, ILIS, and reports numerical evidence that ATAIS outperforms it and standard MCMC benchmarks on several examples.

What carries the argument

The load-bearing object is the alternating fixed-point pair formed by Eq. (18) and Eq. (19): at iteration t, theta_MAP is estimated from a cloud of importance samples targeting the conditional posterior with current covariance Sigma_ML^(t-1), and then Sigma_ML^(t) is set to the empirical outer-product average of the residuals at that $\theta$ estimate. ATAIS also adapts the proposal mean to the current MAP estimate and uses a cyclic scale parameter delta_t for exploration; a final weight correction reweights all particles to the last conditional posterior. The recycling step in the second stage multiplies each particle's conditional weight rho_t^(n)(Sigma) by the prior/proposal ratio gamma_j of a drawn covariance matrix, which is what allows the full posterior and marginals to be assembled from the same $\theta$ samples.

What would settle it

Run complete ATAIS on a two-mode likelihood whose modes have different residual covariance structures and start the algorithm from several initial matrices; if the sequence of Sigma_t cycles between two matrices or the final weighted posterior concentrates on the mode selected by the initial Sigma regardless of the data, the fixed-point assumption fails. A cleaner check is computing the marginal-likelihood estimate (Eq. (51)) from two independent restarts: if the estimates differ by more than their Monte Carlo error, the recycled cloud is not a faithful representation of the full posterior.

Watch

Extended reading notes

Core claim

The central claim is that the alternating scheme converges to the pair (theta_MAP, Sigma_ML) and that the resulting weighted sample cloud represents the conditional posterior p($\theta$ | Y, Sigma_ML) after a final reweighting; drawing Wishart matrices centered at Sigma_ML and reweighting the same cloud yields particle approximations of p($\theta$, Sigma | Y), p($\theta$ | Y), p(Sigma | Y), and the marginal likelihood. The second stage is essentially free because the residuals e_t,r^(n) = y_r - f_r(theta_t^(n)) are saved during the first stage. Across the paper's numerical experiments, this complete procedure provides low mean absolute error in estimating $\theta$ and Sigma, including a 59-dimensional graph-topology problem, and outperforms the benchmark algorithms tested.

Load-bearing premise

The whole scheme assumes that the back-and-forth updates between the best parameter point and the best noise covariance actually settle on the single globally best pair, instead of cycling or locking onto a less good local answer; the paper only argues that using more particles shrinks the error, without proving this convergence.

Editorial extensions

If this is right

Complete ATAIS delivers an approximation of p(theta, Sigma | Y) at the same model-evaluation cost as sampling theta alone, because the covariance stage reuses saved residuals.
Credible intervals for individual covariance entries, such as the 95% intervals reported for the sensor-localization and biological-system examples, become computable by resampling Wishart draws with weights lambda_j.
The method extends to Student-t and other elliptical noises through a fixed-point covariance estimator, and to large datasets via mini-batches, with experiments showing small errors in both settings.
In the 59-dimensional graph-topology example the precision matrix recovered by thresholding the estimated Sigma matches the ground truth adjacency in 69.7% of runs.

Reading between the lines

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

Editorial inference: the recycling step implicitly conditions on the theta-cloud and reweights over Sigma only, so the variance of the resulting posterior estimators should be smaller than sampling the joint space directly, especially when the theta cloud already sits near the mode.
Editorial inference: for large K, the empirical covariance update (Eq. (19)) may be unstable; plugging a shrinkage estimator into the same alternating loop would be a natural, untested variant.
Editorial inference: the Wishart prior location is set to the data-derived Sigma_ML, so for small sample sizes R the reported credible intervals may understate the true uncertainty; a hierarchical prior over the reference matrix would test this.
Editorial inference: running the complete procedure from multiple random starts and comparing the marginal-likelihood estimate (Eq. (51)) would make a practical convergence diagnostic for the alternating fixed-point step.

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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

Quick take: the first stage of this paper is genuinely useful - alternating IS over theta with closed-form ML updates of Sigma, in a gradient-free setting, for nonlinear multioutput models - and the sample-recycling idea in the second stage is practically nice. But the headline claim, that complete ATAIS approximates the joint posterior p(theta,Sigma|Y), is not backed by the analysis as written, and the paper gives no diagnostics to show the second-stage weights are well behaved.

What is actually new: the extension from the authors' earlier scalar-noise ATAIS to a full unknown covariance matrix is non-trivial and useful, and the two-block structure with recycling (no additional evaluations of the expensive model f) is a real practical virtue. The experimental coverage is decent: sensor localization, a multi-output nonlinear model, an ODE biology system, and graph topology estimation with total dimension 59. The MAE tables mostly show expected monotone improvement with N and T, and the method is extended to Student-t noise and mini-batches. The citation pattern is fine; the work is properly anchored to the authors' prior ATAIS papers. Credit is also due for flagging the empirical-Bayes nature of Eq. (59) and for admitting that the convergence argument is heuristic.

Soft spots, in order of size:

1. The joint reweighting in Section 5 is not valid importance sampling as stated. The theta-samples come from proposals tuned to the conditional posterior at Sigma_ML. For a Wishart draw Sigma_j with larger scale or different orientation, nothing guarantees that q(theta|mu_t,Lambda_t) dominates p(theta|Y,Sigma_j), so the weights in Eq. (48) can have large or infinite variance. The paper itself warns in Section 4.4 that the plain IS denominator is unstable and recommends the mixture denominator of Eq. (28) - then uses the plain denominator in Eq. (48). No ESS or weight-variance diagnostics are reported for the joint weights beta or the marginal Sigma weights lambda_j. In the reported credible-interval experiments nu=100, so Wishart draws concentrate near Sigma_ML and the problem may be mild there. But as a generic method, the central claim is not supported. This is addressable: report ESS for the joint weights, use a mixture or defensive denominator, or restrict the claim to settings where the Sigma-proposal is concentrated near the first-stage estimate.

2. Convergence of the alternating fixed-point iteration is asserted, with a rate borrowed from a different sampler; there is no contraction argument. The convergence plots for the test problems look fine, so I read this as a theory gap, not a sign of failure.

3. The graph topology experiment omits key generation details - R, the noise level, the actual Sigma used - and is not reproducible as written.

4. The benchmarks are weak (naive MH variants and a 'helped' MH that uses ATAIS's own Sigma_ML). The no-gradient constraint partially justifies this, but the comparison table should not be read as strong evidence against modern adaptive IS or SMC samplers.

This paper is for practitioners who need a gradient-free sampler for joint parameter/covariance inference in costly nonlinear models. It deserves a serious referee. I would send it out, and require the second-stage diagnostics (or a softened claim) before acceptance.

Referee Report

4 major / 4 minor

Summary. This manuscript introduces adaptive target adaptive importance sampling (ATAIS) for Bayesian inversion problems with an unknown noise covariance. The first stage alternates an importance-sampling approximation of the conditional posterior p(θ|Y,Σ_ML) with a closed-form maximum-likelihood update of Σ given the current particle maximum; after T iterations it outputs weighted θ particles and estimates (θ_MAP,Σ_ML). The second stage draws covariance matrices from a Wishart prior/proposal whose scale is set to Σ_ML/ν, reweights the existing θ particles with joint importance weights, and thereby claims an approximation of the complete posterior p(θ,Σ|Y) with no additional model evaluations. An alternative scheme, ILIS, is described. Numerical experiments cover localization, a multi-output model, mini-batches, Student-t noise, a biology ODE system, graph topology estimation, and comparisons with MCMC benchmarks.

Significance. The recycling idea is attractive: if the second-stage weights were valid, complete Bayesian inference over θ and Σ would be obtained at negligible extra cost relative to the first stage. The numerical experiments show low MAE in several settings and a favorable comparison against MCMC benchmarks with comparable model evaluations. The paper also demonstrates flexibility with non-Gaussian noise and mini-batches. However, the central claim requires stronger theoretical and numerical support than currently given; in particular, the convergence of the alternating optimization and the validity of the recycled importance weights are not established. With those gaps addressed, the method would be a useful contribution.

major comments (4)

[Section 3, Eqs. (18)-(19) and Table 1] Convergence of the alternating optimization is load-bearing and is not demonstrated. The paragraph 'On the convergence of the alternating optimization' asserts that the error can be bounded by increasing N and cites Akyildiz et al. [38] for an O(N^{-1/(2(M+1))}) rate, but [38] analyzes a different sampler and the argument does not show contraction, uniqueness, or global convergence of the fixed-point map (18)-(19). Because each iteration uses the particle maximum θ_max^(t) rather than the exact θ_MAP, the sequence could in principle cycle or converge to a local fixed point; if it does, the final Σ_ML^(T) and every downstream reweighted posterior and credible interval are affected. Please provide a convergence theorem under explicit assumptions, or at minimum a systematic empirical study of restart dependence and path dependence of Σ_ML^(T) with convergence diagnostics, and state the conditions under which the fixed point is global.
[Section 5.2, Eqs. (48)-(49)] The recycled joint weights are not shown to be valid importance weights. The denominator q(θ|μ_t,Λ_t) was adapted to the conditional posterior at Σ_ML, while the numerator targets p(θ|Y,Σ^(j)) for Wishart draws Σ^(j) with potentially larger scale or different orientation. The required domination condition q ≫ p(θ|Y,Σ^(j)) g_θ for all Σ^(j) in the support of q_Σ is neither proved nor checked; if it fails, the weights β have infinite or very large variance. Note that Section 4.4 itself warns that the standard IS denominator produces instability and high variance and recommends a mixture denominator, yet the second stage still uses the standard denominator. Please provide a boundedness/domination argument and report ESS or variance diagnostics for the joint weights β and the marginal weights λ_j.
[Section 5.4, Eq. (59) and Section 5] The prior and proposal for Σ are set to Φ=Σ_ML^(T)/ν, where Σ_ML^(T) is estimated from the same dataset used in the first stage and in the subsequent reweighting. The paper acknowledges this as empirical Bayes, but the reported 95% intervals in Eqs. (67), (70), and (73) are then not calibrated Bayesian posterior intervals; they are conditional on a data-dependent prior location. The claim in Sections 5.2 and 8 that complete ATAIS provides an approximation of the complete posterior should be qualified, and the intervals should be validated by a repeated-sampling coverage study or by a two-stage analysis that accounts for the estimation of Φ.
[Section 7.6] The graph topology experiment omits essential data-generation details, so the reported 69.7% recovery rate is not interpretable. The text does not specify R, the number of observation vectors, the time points τ, the true covariance Σ_true, or the noise level used to generate Y from Eq. (74); it also does not justify the ad hoc threshold 0.3 applied to P=Σ^{-1} or report sensitivity to that threshold. Without these details, and without a baseline method, the experiment cannot support the claim that ATAIS estimates graph topology well.

minor comments (4)

[Throughout] Typographical errors such as 'di fferent', 'Vwhicular Technology' in reference [10], 'Statistical Sicence' in [47], and 'Statistic Computing' in [59] should be corrected.
[Eqs. (21), (26), (28), Table 8] Notation should be made more consistent: 'unit matrix' in Eq. (21) should be 'identity matrix'; the normalized weights are written as wbar in Eq. (26) and wbar^(n)_t in Table 8; and the denominator in Eq. (28) should be defined precisely when the sample is indexed by t.
[Table 5] In the multi-output model, the row with N=100 and T=50 reports a θtrue MAE of 0.009 while neighboring rows report values around 0.001; please check whether this is a typo.
[Section 5.2, Eq. (51)] The marginal likelihood estimator divides by JNT, the product of the number of matrices and all particles; clarify whether N and T include samples discarded by the criterion in Section 4.3, since that criterion suggests saving only a subset of particles.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ATAIS derivation is self-contained, and the acknowledged empirical Bayes prior does not make the posterior equal to the fitted covariance by construction.

full rationale

The derivation chain is self-contained. ATAIS's first stage defines θMAP and ΣML as separate solutions of Eqs. (13)-(14) and (18)-(19); the alternating iteration is a fixed-point search, not a definitional identity, and the IS weights in Eqs. (76) and (79) are ordinary target/proposal ratios. The final correction π_{T+1}/π_t is a likelihood-ratio reweighting with no fitted quantity renamed as a prediction. The second stage is a factored-proposal IS identity: Eq. (48) targets p(θ,Σ|Y) using q(θ|µ_t,Λ_t)qΣ(Σ), and recycling θ samples is valid whenever the support condition holds. The absence of a dominance proof for all Σ_j in the Wishart support is a robustness gap, not a circular reduction. The empirical Bayes choice Φ = ΣML^(T)/ν in Eq. (59) is explicitly labeled as empirical Bayes; with qΣ = gΣ, the prior enters through the sampling distribution of the Σ_j rather than as an enforced equality, and the reported posterior is a weighted average of drawn matrices, not an identity with ΣML. The self-citations ([7,15,16,22,23,26,27]) supply context, terminology, and standard IS/Gibbs concepts; none is load-bearing for the correctness of the central algorithm, and no uniqueness theorem is imported from the authors' prior work. The unproved convergence of the alternating fixed-point map is an omitted support gap, not a circular step. External checks in Section 7.7 and MAE evaluations against θtrue and ΣML ground truths provide independent evidence. Therefore no circular step is exhibited, and the appropriate score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central algorithm rests on user-chosen adaptation schedules, an empirical Bayes covariance prior centered at the first-stage fit, and several unproved convergence and consistency assumptions. No new physical or mathematical entities are postulated.

free parameters (4)

Wishart degrees of freedom ν for the Σ prior/proposal = ν = 100 in credible interval experiments; ν = 4 or 5 for ILIS
Controls how tightly the proposed covariance matrices are concentrated around the data-derived Φ. The large ν = 100 value makes the reported Σ intervals strongly reflect the empirical Bayes prior.
Empirical Bayes prior/proposal scale Φ = Σ_ML^(T)/ν = Set to the final ATAIS ML covariance estimate
Chosen from the same data used for inference, so the second-stage posterior over Σ inherits the first-stage fitted covariance.
Cyclic covariance inflation parameters δ0, δmin, a = δ0 = 1, δmin = 0.05, a = 0.1 in the graph topology example
User-chosen schedule controlling the exploration/exploitation balance of the proposal covariance; no automatic selection rule is provided.
Optional discard threshold ε = 0.3 (recommended)
Used to decide which early proposals to discard in the optional multiple importance sampling weighting; chosen by empirical experience.

assumptions (5)

domain assumption Every conditional posterior π_t has a normalized proposal q_t with heavier tails and finite importance weight variance.
Required for the IS weights in Eqs. (76) and (79) to be well behaved; the paper notes this requirement but does not verify it for the numerical targets.
ad hoc to paper The alternating iteration between θMAP (Eq. 18) and ΣML (Eq. 19) converges to the global pair (θMAP, ΣML), not to a cycle or a local fixed point.
The convergence section in Section 3 is informal and cites a rate for a different sampler; no proof is provided for the specific alternating map used here.
domain assumption The standard single-proposal IS weights in Table 8 remain consistent even though the proposals are adapted from previous samples.
Section 4.4 acknowledges that adaptive IS can be unstable with the standard denominator and offers an optional multiple IS denominator, but the main algorithm uses the standard weights.
standard math For Student-t noise, the fixed-point covariance estimator in Section 3.2 is consistent.
The paper relies on conditions from reference [43] for existence and uniqueness of the scale matrix estimator; this is reasonable but not verified in the paper.
ad hoc to paper Thresholding entries of the estimated precision matrix P = Σ^{-1} at 0.3 recovers the true graph adjacency matrix.
Section 7.6 applies this threshold without sensitivity analysis, and the graph experiment does not state how the data or noise covariance were generated.

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Pith. "Pith review of Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals." pith.science (2026). https://pith.science/paper/RTURVDWB

@misc{pith2026250101148,
  author       = {Pith},
  title        = {Pith review of: Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTURVDWB}},
  note         = {Machine review of arXiv:2501.01148}
}

read the original abstract

In this paper we address the problem of performing Bayesian inference for the parameters of a nonlinear multi-output model and the covariance matrix of the different output signals. We propose an adaptive importance sampling (AIS) scheme for multivariate Bayesian inversion problems, which is based in two main ideas: the variables of interest are split in two blocks and the inference takes advantage of known analytical optimization formulas. We estimate both the unknown parameters of the multivariate non-linear model and the covariance matrix of the noise. In the first part of the proposed inference scheme, a novel AIS technique called adaptive target adaptive importance sampling (ATAIS) is designed, which alternates iteratively between an IS technique over the parameters of the non-linear model and a frequentist approach for the covariance matrix of the noise. In the second part of the proposed inference scheme, a prior density over the covariance matrix is considered and the cloud of samples obtained by ATAIS are recycled and re-weighted to obtain a complete Bayesian study over the model parameters and covariance matrix. ATAIS is the main contribution of the work. Additionally, the inverted layered importance sampling (ILIS) is presented as a possible compelling algorithm (but based on a conceptually simpler idea). Different numerical examples show the benefits of the proposed approaches

Figures

Figures reproduced from arXiv: 2501.01148 by the authors.

**Figure 2.** Location Example. MAE in the estimation of [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Location example. Convergence of the components in [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Location example: histogram of the components of [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]

**Figure 5.** Figure 5: Location example: Estimations of θMAP with different schemes. the green squares represent the estimation with ATAIS; the red circles depict for the estimations of ILIS with a covariance matrix generated from a Wishart distribution. The blue diamonds represent the estim…

**Figure 6.** Figure 6: Multi-output example. Convergence of the components in the estimators [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]

**Figure 7.** Figure 7: Multi-output example. MAE in the estimation of the MAP of the posterior (considering [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]

**Figure 8.** Figure 8: Multioutput example. (a) Complete MAE in estimating the θMAP and the ΣML, keeping fixed the number of posterior evaluations NT = 20000, with T ∈ {100, 200, 500, 1000, 2000}, hence N ∈ {200, 100, 40, 20, 10}. (b) Estimations of θMAP. The green squares represent the esti…

**Figure 9.** Figure 9: Testing the use of mini-batches in the multi-output model in Section 7.2: MAE in estimating [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]

**Figure 10.** Figure 10: Application to a system biology example. Evolution of the components in the estimators [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]

**Figure 11.** Figure 11: Graph topology estimation with ATAIS algorithm (with [PITH_FULL_IMAGE:figures/full_fig_p037_11.png]

**Figure 12.** Figure 12: (a) Evaluations of the final conditional log-posterior of the samples θ (n) t such that w¯ (n) t ≥ 1/N at each iteration. (b) Histograms of the four components of the relevant samples. They are localized around the true values of θtrue = [0.5, 2, 5, 3]⊤ . 7.7 Comparis…

Discussion (0). Continue with ORCID to comment.

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Recall πt(θ)∝ p(θ|Y, ˆ︁Σ(t−1) ML )

Initializations: Choose N, µ1, Λ1, ˆ︁Σ(0) ML, and setπMAP = 0. Recall πt(θ)∝ p(θ|Y, ˆ︁Σ(t−1) ML )
[69]

Draw θ(1) t ,..., θ(N) t ∼ q(θ|µt, Λt)

For t = 1,..., T: (a) Sampling: i. Draw θ(1) t ,..., θ(N) t ∼ q(θ|µt, Λt). ii. Assign to each sample the weights w(n) t = πt(θ(n) t ) q(θ(n) t |µt, Λt) , n = 1,..., N. (76) (b) Current maximum estimations: i. Obtain θ(t) max = arg max n πt(θ(n) t ), and computeˆ︁rt = fr(θ(t) m...
[70]

Output: Return the final estimators ˆ︁θ(T) MAP, ˆ︁Σ(T) ML , and all the weighted samples {θ(n) t , ˜︁w(n) t }, for all t and n, with the corrected weights ˜︁w(n) t = w(n) t πT+1(θ(n) t ) πt(θ(n) t ) . (79) 45

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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