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arxiv: 2605.00332 · v1 · submitted 2026-05-01 · 📊 stat.ME · cs.NA· math.NA

Beyond Independence: on Jointly Normal Priors in Bayesian Inversion

Ruanui Nicholson , Matti Niskanen , Oliver J. Maclaren , Jari P. Kaipio This is my paper

Pith reviewed 2026-05-09 19:47 UTC · model grok-4.3

classification 📊 stat.ME cs.NAmath.NA
keywords joint Gaussian priorsBayesian inversionmarginal preservationcross-correlationcovariance constructionstrict contractionuncertainty in correlation
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The pith

Jointly normal priors can include correlations between parameters while exactly preserving each parameter's marginal distribution.

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

The paper develops a construction for jointly Gaussian prior models in Bayesian inversion problems that match any prescribed Gaussian marginal distributions for each unknown parameter. Correlations are encoded through strict contraction factors, which permits spatially varying cross-correlations and allows the correlation strength itself to be treated as uncertain and inferred from data. Standard approaches either assume the parameters are independent or impose similarity through regularization whose statistical interpretation is unclear. A sympathetic reader would care because the method directly addresses how neglecting or fixing correlations can distort posterior uncertainty and inference in multi-parameter inverse problems, while keeping all unknowns as random variables.

Core claim

We construct jointly Gaussian prior models with prescribed Gaussian marginals so that correlation between the parameters can be incorporated without altering the marginal prior distributions. We propose a joint covariance construction that preserves the marginals, allows spatially varying cross-correlation, and supports uncertainty and inference in the correlation itself. The construction is valid for any strict contraction encoding the desired cross-correlation and is optimal in a canonical correlation sense under the principal square root factorisation. Demonstrations via prior sampling and inference examples, including a PDE-constrained problem, highlight pitfalls of ignoring correlation,

What carries the argument

The joint covariance construction that uses strict contraction factors to encode cross-correlations while preserving the given marginal distributions.

If this is right

  • Spatially varying cross-correlations can be included without changing the marginal prior distributions.
  • The strength of the correlation can be treated as a random variable and inferred jointly with the parameters.
  • The construction remains valid for any strict contraction that encodes the desired cross-correlation.
  • Neglecting uncertainty in the correlation produces incorrect posterior uncertainty estimates in the examples.

Where Pith is reading between the lines

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

  • Routine inclusion of correlation as an inferred quantity would reduce bias in uncertainty estimates for many joint inversion tasks.
  • The optimality property under principal square root factorization could guide choices among different matrix factorizations in similar constructions.

Load-bearing premise

The desired cross-correlations between parameters can always be represented by a strict contraction factor without producing invalid non-positive-definite joint covariances or violating the prescribed marginal distributions.

What would settle it

A concrete calculation for some choice of marginal variances and contraction factor that yields a joint covariance matrix with a negative eigenvalue would disprove the claim that the construction is always valid.

Figures

Figures reproduced from arXiv: 2605.00332 by Jari P. Kaipio, Matti Niskanen, Oliver J. Maclaren, Ruanui Nicholson.

Figure 1
Figure 1. Figure 1: Samples from the joint priors in Example 1. The first row shows three samples of p, the second row shows three corresponding samples of m, where the desired correlation structure is encoded using c(x) = 0.999 (Example 1 (a)), and the third row shows three corresponding samples of m, where the desired correlation is encoded using c(x) = 0.999 for x ≤ 1 and c(x) = −0.999 for x > 1 (Example 1 (b)). As expecte… view at source ↗
Figure 2
Figure 2. Figure 2: Three samples from the joint priors in Example 2. In the top row we show three samples of p, while on the bottom row we show the three corresponding samples of m in red as well as the restriction of p to the boundary, i.e., p|Bb , in blue. The desired correlation structure is encoded using c(x) = 0.999 for x ∈ Bb. In view at source ↗
Figure 3
Figure 3. Figure 3: Demonstration of the effects of different factorisations of Γp and Γm. On the left we show the desired correlation structure encoded in diag(C) (in black), along with the actual resulting diagonal of the correlation matrices Φ⋆ (in blue) and Φ▷ (in red) found using the principal square root and the Cholesky factor, respectively. In the centre and on the right we show samples from the joint prior of p (in b… view at source ↗
Figure 4
Figure 4. Figure 4: Joint distributions for the Monod example with fixed values of c. On the left are the joint priors, in the middle are the joint posteriors for δe = 0.1, while on the right are the joint posteriors for δe = 0.03. In all cases the colors red, yellow, black, green, blue are used for the cases c = −0.99, c = −0.85, c = 0, c = 0.85, and c = 0.99, respectively. Furthermore, in all cases the black cross represent… view at source ↗
Figure 5
Figure 5. Figure 5: Corner plot for the posterior of the Monod example where c is also treated as an unknown and estimated. The true parameters are (ptrue, mtrue) = (0.7, 65). 4. Higher-dimensional computational experiments. In this section we consider two examples motivated by geophysical situations in which the (discretised) parameters are significantly higher dimensional than the previous example. We first consider an exam… view at source ↗
Figure 6
Figure 6. Figure 6: The true parameters ptrue (left) and mtrue (right). In each plot the black dots indicate the locations in which the noisy (direct) point-wise measurements are taken. 4.2. Computational details for Example 1. In this example we parameterise the unknown correlation using C = cI, where c is a scalar with a uniform prior distribution, i.e., c ∼ U(−1, 1). That is, the correlation is assumed to be homogeneous th… view at source ↗
Figure 7
Figure 7. Figure 7: True parameters and CM estimates for Example 1. In the top row we show the CM found using independent inference (left), the true parameter ptrue (centre), and the CM found using joint inference (right). On the bottom row the corresponding plots for m. p m view at source ↗
Figure 8
Figure 8. Figure 8: The reduction in uncertainty for Example 1. On the top row we show the point-wise marginal posterior standard deviation for p found using independent inference (left), the point-wise marginal posterior standard deviation for p found using joint inference (centre), and the difference of the two Dp (see Equation 4.3). On the bottom row we show the equivalent figures for m. approach the CM estimates improve s… view at source ↗
Figure 9
Figure 9. Figure 9: Results for estimating the correlation c in Example 1. Shown on the left is the histogram of samples from the marginal posterior π(c|d), the prior π(c) (red line), and the truth ctrue (black dashed line), while in the centre we show the trace of the samples for c and on the right the sample autocorrelation. A key benefit of the proposed joint approach is the ability to estimate the correlation also. In view at source ↗
Figure 10
Figure 10. Figure 10: Results for Example 1 using fixed values of c. On the top row we show the CM estimate of p using c = ctrue = −0.9 (left), the CM estimate of p using c = −ctrue = 0.9 (centre), and the point-wise marginal posterior standard deviation of p using c = ctrue or c = −ctrue (right). Recall the point-wise marginal posterior standard deviation is invariant to the sign of c. In the bottom row we show the correspond… view at source ↗
Figure 11
Figure 11. Figure 11: True parameters and data for Example 2. The true log-permeability ptrue (left), the true log-recharge mtrue (centre), as well as the resulting hydraulic-head u (right). In each plot the black dots indicate the locations of the measurements. 4.4. Computational example 2: Inversion for aquifer permeability and recharge. We now consider the more realistic subsurface flow problem of estimating the permeabilit… view at source ↗
Figure 12
Figure 12. Figure 12: Laplace approximation used to ‘warm-start’ the MCMC sampling for Example 2. In the top row we show the true log-permeability ptrue (left), the MAP estimate (centre) and the approximate point-wise marginal posterior standard deviation (right). p m view at source ↗
Figure 13
Figure 13. Figure 13: True parameters and CM estimates for Example 2. In the top row we show the CM found using independent inference (left), the true log-permeability ptrue (centre), and the CM found using joint inference (right). On the bottom row we show the corresponsing plots for the log-recharge m. reconstructions of both p and m, particularly for the recharge field m at the well locations (where p was measured directly). In view at source ↗
Figure 14
Figure 14. Figure 14: The reduction in uncertainty for Example 2. On the top we show the point-wise marginal posterior standard deviation for p found using independent inference (left), the point-wise marginal posterior standard deviation for p found using joint inference (centre), and the difference of the two, Dp (right). On the bottom row we show the corresponding plots for m view at source ↗
Figure 15
Figure 15. Figure 15: Results for estimating the correlation in Example 2. We show the joint marginal posterior π(c1, c2|d) (left) as well as the marginal posteriors of π(c1|d) (centre) and π(c2|d) (right). In each of the plots the red line is used to indicate the (support) of the relevant prior distribution, while the black dot in the joint marginal plot (right) and black dotted lines in the marginal plots shows the true valu… view at source ↗
Figure 16
Figure 16. Figure 16: MCMC diagnostics for the correlation parameters c1 and c2 in Example 2. The left and centre panels show the trace plots for c1 and c2, respectively, while on the right we show the corresponding sample autocorrelation functions. Independent Joint view at source ↗
Figure 17
Figure 17. Figure 17: MCMC diagnostics for the correlation parameters for the KL modes pˆ and mˆ in Example 2 using independent and joint inference. On the top row we show the trace plot of the first KL mode for the log-permeability pˆ1 using independent inversion (left), the trace plot of the first KL mode for the log-recharge mˆ 1 using independent inversion and the sample autocorrelation for all KL modes using independent i… view at source ↗
read the original abstract

We consider joint inversion for two or more unknown parameters from observational data in the Bayesian framework. Standard approaches often either treat the parameters as independent or impose structural similarity through regularisation terms that can be difficult to interpret statistically. We instead construct jointly Gaussian prior models with prescribed Gaussian marginals, so that correlation between the parameters can be incorporated without altering the marginal prior distributions. We propose a joint covariance construction that preserves the marginals, allows spatially varying cross-correlation, and supports uncertainty and inference in the correlation itself. The construction is valid for any strict contraction encoding the desired cross-correlation and is optimal in a canonical correlation sense under the principal square root factorisation. We demonstrate the method using prior sampling and several inference examples: a low-dimensional illustrative example and two higher-dimensional examples, including a PDE-constrained problem. The examples highlight both the potential pitfalls of ignoring or neglecting uncertainty in the correlation as well as reinforcing a key principle of the Bayesian paradigm: unknown quantities included in a model should be treated as random variables.

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

2 major / 2 minor

Summary. The paper proposes a construction for jointly Gaussian prior models in Bayesian inversion of multiple parameters. It preserves prescribed Gaussian marginal distributions while incorporating cross-correlations via any strict contraction factor, with the joint covariance claimed to be valid for any such contraction and optimal in a canonical correlation sense under the principal square root factorization. The approach is illustrated through prior sampling and inference examples, including a low-dimensional case, higher-dimensional cases, and a PDE-constrained problem, with emphasis on treating correlation uncertainty as random.

Significance. If the central construction holds, the work offers a statistically grounded alternative to independence assumptions or ad-hoc regularization in multi-parameter Bayesian inversions. It enables spatially varying correlations without distorting marginal priors and supports inference on the correlation structure itself. The examples demonstrate practical value in uncertainty quantification, particularly for PDE-constrained settings, and align with core Bayesian principles by randomizing unknown quantities like correlations.

major comments (2)
  1. [Construction and theoretical claims (likely §3 or equivalent)] The abstract and introduction assert validity for any strict contraction and optimality under principal square root factorization, but the manuscript does not provide the explicit matrix construction or step-by-step verification that the resulting joint covariance remains positive definite while exactly preserving the marginal covariances. This verification is load-bearing for the central claim; without it, the support for the proposal cannot be fully assessed from the given examples alone.
  2. [Examples and PDE-constrained demonstration] The weakest assumption—that any desired cross-correlation can be encoded by a strict contraction without producing non-positive-definite joints—is stated but not accompanied by a general proof or counterexample analysis. The low-dimensional example may satisfy it by construction, but this needs explicit confirmation for the higher-dimensional and PDE cases to substantiate the broad applicability.
minor comments (2)
  1. [Notation and setup] Notation for the contraction factor and factorization could be introduced with a dedicated table or diagram early in the paper to improve readability for readers unfamiliar with canonical correlation analysis.
  2. [Numerical results] The discussion of pitfalls when ignoring correlation uncertainty would be strengthened by a direct quantitative comparison (e.g., posterior variance or credible interval width) between the proposed joint model and an independence baseline in one of the numerical examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report, which highlights both the potential value of the proposed joint prior construction and areas where the presentation can be strengthened. We address each major comment below and will incorporate the requested clarifications and verifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Construction and theoretical claims (likely §3 or equivalent)] The abstract and introduction assert validity for any strict contraction and optimality under principal square root factorization, but the manuscript does not provide the explicit matrix construction or step-by-step verification that the resulting joint covariance remains positive definite while exactly preserving the marginal covariances. This verification is load-bearing for the central claim; without it, the support for the proposal cannot be fully assessed from the given examples alone.

    Authors: We agree that the central claim requires explicit support. The manuscript defines the joint covariance via the principal square-root factorization of the marginal covariances combined with a contraction operator, but we acknowledge that a self-contained derivation of positive definiteness and exact marginal preservation is not fully expanded. In the revision we will add, in the theoretical section, the explicit block-matrix expression together with a short proof: the Schur complement of the joint matrix reduces to a positive-definite term precisely when the contraction has spectral radius strictly less than one, while the diagonal blocks recover the prescribed marginal covariances by construction. This will be accompanied by the canonical-correlation optimality argument under the principal-square-root choice. revision: yes

  2. Referee: [Examples and PDE-constrained demonstration] The weakest assumption—that any desired cross-correlation can be encoded by a strict contraction without producing non-positive-definite joints—is stated but not accompanied by a general proof or counterexample analysis. The low-dimensional example may satisfy it by construction, but this needs explicit confirmation for the higher-dimensional and PDE cases to substantiate the broad applicability.

    Authors: We accept that a general statement without supporting analysis is insufficient. The low-dimensional example is verified directly, but the higher-dimensional and PDE examples rely on the same contraction mechanism without separate confirmation. In the revision we will insert a concise general lemma establishing that any strict contraction (operator norm <1) yields a positive-definite joint while leaving marginals unchanged, together with a short remark on the boundary case of non-strict contractions. We will also add a brief numerical check of the smallest eigenvalue of the joint covariance in the two higher-dimensional examples (including the PDE-constrained case) to illustrate that the property holds in practice. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct mathematical construction for jointly Gaussian priors that preserves prescribed marginals via any strict contraction factor and principal square root factorization. This is an explicit definition of the joint covariance (valid by the contraction property and optimality under canonical correlation), not a reduction of a target quantity to fitted inputs, self-referential equations, or load-bearing self-citations. The examples illustrate the construction without deriving the core result from the data or prior outputs. The derivation chain is self-contained as a proposal of a new prior model.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that parameters admit jointly Gaussian priors with user-specified Gaussian marginals, plus the mathematical claim that any strict contraction yields a valid joint covariance under the principal square root factorization.

free parameters (1)
  • contraction factor
    Encodes the desired cross-correlation strength and may be chosen by the user or inferred from data.
axioms (2)
  • domain assumption Parameters follow jointly Gaussian distributions with prescribed Gaussian marginals
    The entire construction is defined for jointly normal priors that keep the given marginal distributions unchanged.
  • domain assumption A strict contraction can represent any desired cross-correlation structure
    Validity and optimality claims are stated to hold for any such contraction.

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discussion (0)

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Reference graph

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