Local sensitivity analysis for Bayesian inverse problems

David Ebert; J\"urgen D\"olz

arxiv: 2503.20526 · v3 · submitted 2025-03-26 · 🧮 math.NA · cs.NA

Local sensitivity analysis for Bayesian inverse problems

J\"urgen D\"olz , David Ebert This is my paper

Pith reviewed 2026-05-22 22:39 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Bayesian inverse problemslocal sensitivity analysisasymptotic expansionsposterior momentsuncertainty quantificationperturbation approachnumerical experiments

0 comments

The pith

Moments of random variables under the posterior can be approximated by asymptotic expansions in Bayesian inverse problems.

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

The paper extends the perturbation approach from local sensitivity analysis to Bayesian inverse problems. It shows that posterior moments of random variables can be obtained via asymptotic expansions when the measurement operators and prediction functions meet smoothness conditions and possess finite moments under the prior. A sympathetic reader would care because this supplies an efficient route to uncertainty quantification that avoids repeated sampling of the full posterior. The work includes numerical experiments that demonstrate the expansions in practice.

Core claim

Under the assumption that measurement operators and prediction functions are sufficiently smooth and that the corresponding stochastic moments with respect to the prior distribution exist, moments of random variables with respect to the posterior distribution can be approximated efficiently by asymptotic expansions obtained from the local sensitivity analysis or perturbation approach.

What carries the argument

Asymptotic expansions derived from the perturbation approach (local sensitivity analysis) applied to posterior moments.

If this is right

Posterior moments become computable without drawing large numbers of samples from the posterior.
Uncertainty quantification for Bayesian inverse problems can be performed at lower computational cost when the smoothness assumptions hold.
The accuracy of the moment approximations increases with the order of the expansion under the stated conditions.
Numerical experiments confirm that the expansions reproduce reference values in chosen test problems.

Where Pith is reading between the lines

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

The approach may scale to higher-dimensional inverse problems where full posterior sampling becomes prohibitive.
Similar expansions could be explored for other summary statistics beyond moments, such as quantiles.
One could test the method by comparing run times against standard sampling techniques on standard benchmark inverse problems.

Load-bearing premise

The measurement operators and prediction functions must be sufficiently smooth and their stochastic moments with respect to the prior distribution must exist.

What would settle it

In a low-dimensional test case where the smoothness and moment conditions hold, compute the exact posterior moments by direct integration and check whether the asymptotic expansion matches those values to the expected order.

read the original abstract

We present an extension of local sensitivity analysis, also referred to as the perturbation approach for uncertainty quantification, to Bayesian inverse problems. More precisely, we show how moments of random variables with respect to the posterior distribution can be approximated efficiently by asymptotic expansions. This is under the assumption that the measurement operators and prediction functions are sufficiently smooth and their corresponding stochastic moments with respect to the prior distribution exist. Numerical experiments are presented to the illustrate the theoretical results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper applies standard asymptotic perturbation methods to approximate posterior moments in Bayesian inverse problems under explicit smoothness assumptions.

read the letter

The main takeaway is that the authors extend local sensitivity analysis to Bayesian inverse problems by using asymptotic expansions to approximate posterior moments. They do this under the assumptions that the measurement operators and prediction functions are smooth enough and that the relevant prior moments exist. Numerical experiments are included to show the approach in action. This is a direct but clean application of perturbation ideas to the posterior setting, and the abstract states the hypotheses without overclaiming. The work is honest about its scope as local analysis. The soft spots are that the central argument is a standard perturbation expansion once the hypotheses are granted, so the novelty is mostly in the transfer to this context rather than new technical machinery. The abstract gives no explicit error bounds or rates, which makes it harder to judge practical usefulness from the given material alone. Literature comparisons are not visible here, so it is unclear how this sits relative to other asymptotic treatments of inverse problems. This paper is for applied mathematicians and engineers working on uncertainty quantification in inverse problems who might want a cheap local approximation instead of full posterior sampling. A reader focused on that niche could extract some value from the examples and the stated conditions. It deserves a serious referee because the claim is well-posed, the assumptions are transparent, and the numerical support is at least attempted, even if the overall advance is incremental.

Referee Report

0 major / 2 minor

Summary. The manuscript extends local sensitivity analysis (perturbation approach) to Bayesian inverse problems. It claims that moments of random variables with respect to the posterior distribution admit efficient approximation via asymptotic expansions, provided the measurement operators and prediction functions are sufficiently smooth and the corresponding stochastic moments with respect to the prior exist. Numerical experiments are presented to illustrate the theoretical results.

Significance. If the derivations hold, the work supplies a perturbation-based route to posterior-moment computation that avoids full sampling, which is potentially useful for uncertainty quantification when the forward map is smooth and prior moments are available. The explicit listing of hypotheses is a positive feature; the approach appears to be a direct transfer of standard asymptotic techniques rather than an entirely new framework.

minor comments (2)

The abstract states the main result and assumptions but does not indicate the order of the expansion or the precise form of the leading correction term; adding one sentence on this point would improve clarity for readers.
Numerical experiments are described only as 'illustrative'; a brief statement of the test problems, observed convergence rates, and comparison to sampling methods would strengthen the validation section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review of the manuscript and for recommending minor revision. The report provides a concise summary of the work and notes its potential utility but does not list any specific major comments requiring response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies standard asymptotic expansion techniques to approximate posterior moments in Bayesian inverse problems, under explicitly stated assumptions of sufficient smoothness of measurement operators/prediction functions and existence of prior moments. This is a direct perturbation argument with no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations or uniqueness theorems. The derivation chain is self-contained as a mathematical extension of existing local sensitivity methods, with numerical experiments serving only as illustration rather than input to the claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from analysis; no free parameters or invented entities are mentioned.

axioms (2)

domain assumption Measurement operators and prediction functions are sufficiently smooth
Stated in the abstract as required for the asymptotic expansions to hold.
domain assumption Stochastic moments with respect to the prior distribution exist
Stated in the abstract as a necessary condition for the approximations.

pith-pipeline@v0.9.0 · 5588 in / 1065 out tokens · 61094 ms · 2026-05-22T22:39:49.445229+00:00 · methodology

Local sensitivity analysis for Bayesian inverse problems

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)