Beyond Independence: on Jointly Normal Priors in Bayesian Inversion

TL;DR

This paper introduces a method for constructing jointly Gaussian priors with prescribed marginals and spatially varying correlations, enhancing Bayesian joint inversion by explicitly modeling correlation uncertainty.

Contribution

It proposes a novel covariance construction that preserves marginals, allows spatially varying correlations, and supports inference on correlation uncertainty.

Findings

Demonstrates the method with prior sampling and inference examples

Highlights the importance of modeling correlation uncertainty in Bayesian inversion

Shows potential pitfalls of neglecting correlation uncertainty

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.