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arxiv: 2606.09594 · v1 · pith:KLT5LVSFnew · submitted 2026-06-08 · 🧮 math.ST · cond-mat.stat-mech· cs.NA· math-ph· math.MP· math.NA· stat.TH

Constraint residuals, graph posteriors, and determinant-corrected full-space targets in Bayesian inverse problems

Jonathon Cottom , Emilia Olsson This is my paper

Pith reviewed 2026-06-27 14:39 UTC · model grok-4.3

classification 🧮 math.ST cond-mat.stat-mechcs.NAmath-phmath.MPmath.NAstat.TH
keywords Bayesian inverse problemsconstrained samplingresidual penaltiesdeterminant correctionsgraph posteriorsstate equationsposterior equivalence
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The pith

Residual penalties in full-space Bayesian inverse problems converge to a posterior differing from the reduced graph posterior by a determinant factor unless corrected.

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

The paper shows that sampling equality-constrained Bayesian inverse problems in full parameter-state space via residual penalties does not automatically recover the same posterior measure as sampling in the reduced space where the state is eliminated by the constraint. For finite-dimensional discretizations where the state equation has a unique solution u = G(θ) and nonsingular Jacobian, the reduced posterior, its graph lift, and the zero-noise residual posterior are distinct. A local change of variables demonstrates that an uncorrected Gaussian residual penalty, after marginalization over the state, equals the reduced density multiplied by the inverse absolute determinant of the state Jacobian. The authors derive explicit determinant corrections for unweighted, weighted, and rescaled residuals so that their hard-constraint limits match the graph-lifted reduced posterior. This separates the question of feasibility from the calibration of the posterior density.

Core claim

In finite-dimensional equality-constrained Bayesian inverse problems, the reduced posterior on parameters θ, its graph lift to the manifold u = G(θ), and the zero-noise limit of a residual-penalized full-space posterior are distinct measures. An uncorrected Gaussian residual penalty converges after marginalization over u to the reduced posterior density multiplied by |det D_u c(θ, G(θ))|^{-1}. Determinant corrections applied to unweighted, weighted, and rescaled residual penalties recover the graph-lifted reduced posterior as the hard-constraint limit.

What carries the argument

The Jacobian determinant |det D_u c(θ, G(θ))| arising from the local change of variables between full-space (θ, u) coordinates and the reduced parameter space.

If this is right

  • Algebraically equivalent residual formulations can share the same feasible set yet induce different limiting posteriors.
  • Driving the residual to zero is not sufficient to sample exactly from the graph-lifted reduced posterior without a matching density correction.
  • Corrected residual penalties allow full-space samplers to target the desired reduced posterior in the hard-constraint limit.

Where Pith is reading between the lines

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

  • Existing full-space MCMC methods that rely on residual penalties may be sampling from a measure that differs from the intended reduced posterior by a state-Jacobian factor.
  • The separation of feasibility from density calibration suggests that similar determinant adjustments could be needed when residual penalties are used inside optimization-based or variational inference schemes.

Load-bearing premise

The state equation has a unique solution u = G(θ) for each parameter value together with a nonsingular state Jacobian.

What would settle it

For a low-dimensional test problem with an analytically known reduced posterior, compare the marginal obtained from an uncorrected residual-penalized sampler against the true reduced density and check whether they differ by exactly the predicted determinant factor.

Figures

Figures reproduced from arXiv: 2606.09594 by Emilia Olsson, Jonathon Cottom.

Figure 1
Figure 1. Figure 1: Analytic scalar example and software-level residual-scaling diagnostic. (a) Scaling [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Section 6 validation-scale benchmark generated from the detcorr implementation. The corrected full-space finite-penalty marginal is visually indistinguishable from the reduced-space reference, as predicted by Theorem 1, while the naive full-space marginal is biased by the missing state-Jacobian volume factor [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
read the original abstract

Bayesian inverse problems constrained by state equations are often sampled in a full parameter-state space by penalising the residual, rather than in a reduced space where the state is eliminated. We show that these formulations are not automatically equivalent as posterior measures. For finite-dimensional discretisations of equality-constrained inverse problems, assume the state equation \(c(\theta,u)=0\) has a unique solution \(u=G(\theta)\) and nonsingular state Jacobian \(\D_u c\). The reduced posterior, its graph lift, and the zero-noise residual posterior are then distinct. A local change of variables shows that an uncorrected Gaussian residual penalty converges, after marginalisation over \(u\), to the reduced density multiplied by \(\abs{\det \D_u c(\theta,G(\theta))}^{-1}\). Thus algebraically equivalent residuals can define the same feasible set but different limiting posteriors. We derive determinant corrections for unweighted, weighted, and rescaled residual penalties that have the graph-lifted reduced posterior as their hard-constraint limit. The result separates feasibility from posterior calibration: driving the residual to zero is not sufficient for exact sampling of the graph-lifted reduced posterior unless the sampling or correction step targets the corresponding corrected density.

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.

Referee Report

0 major / 2 minor

Summary. The paper shows that, for finite-dimensional discretizations of equality-constrained Bayesian inverse problems where the state equation c(θ,u)=0 admits a unique solution u=G(θ) with nonsingular Jacobian D_u c, the reduced posterior, its graph lift, and the zero-noise residual posterior in the full (θ,u) space are distinct measures. A local change-of-variables argument establishes that an uncorrected Gaussian residual penalty, after marginalization over u, converges to the reduced density multiplied by |det D_u c(θ,G(θ))|^{-1}. Determinant corrections are derived for unweighted, weighted, and rescaled residual penalties so that their hard-constraint limits recover the graph-lifted reduced posterior. The central message is that algebraic feasibility (zero residual) does not automatically yield the correct posterior calibration.

Significance. If the change-of-variables derivation holds, the result is significant for the theory and practice of constrained Bayesian inverse problems: it rigorously separates the algebraic constraint set from the measure-theoretic target and supplies explicit corrections that can be implemented in full-space sampling algorithms. The argument relies only on the implicit-function theorem and standard Lebesgue-measure transformation, which are standard tools and therefore reproducible; the explicit determinant factors constitute falsifiable predictions for the limiting behavior of residual-based samplers.

minor comments (2)
  1. §2: the definition of the graph lift could be stated as an explicit push-forward measure rather than left implicit, to make the distinction from the reduced posterior immediate.
  2. The abstract and introduction both use the phrase 'graph-lifted reduced posterior'; a single sentence in §1 that equates this object to the push-forward of the reduced density under θ ↦ (θ,G(θ)) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the accurate summary of our results and for the positive recommendation to accept the manuscript. The referee's assessment correctly identifies the central distinction between algebraic feasibility and measure-theoretic calibration that the paper establishes.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard change-of-variables and implicit-function theorem

full rationale

The paper's central result follows directly from the implicit function theorem (unique u = G(θ) with nonsingular D_u c) and the standard transformation rule for Lebesgue measure under a local diffeomorphism. The factor |det D_u c(θ, G(θ))|^{-1} appears as the Jacobian determinant of the graph map, which is an immediate algebraic consequence of the assumed nonsingularity and is not obtained by fitting, self-definition, or any self-citation chain. The separation between the zero-residual feasible set and the corrected posterior measure is likewise a direct consequence of marginalization over the state variable; no step reduces the claimed limit to an input quantity by construction. The argument is scoped to finite-dimensional discretizations and invokes only textbook measure-theoretic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions about the state equation that are stated explicitly in the abstract; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The state equation c(θ,u)=0 has a unique solution u=G(θ)
    Invoked to define the reduced posterior and the graph lift in the finite-dimensional setting.
  • domain assumption The state Jacobian D_u c is nonsingular
    Required for the determinant to be defined and for the local change of variables to be valid.

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