🤖 AI Summary
This work addresses the limitations in multiparameter Bayesian inversion arising from oversimplified assumptions of parameter independence or regularization strategies lacking statistical justification. It proposes a joint Gaussian prior construction that preserves prescribed marginal Gaussian distributions while incorporating spatially varying cross-correlation structures via a principal square-root covariance decomposition. This formulation is optimal in the sense of canonical correlation analysis and enables explicit quantification of uncertainty in the correlation structure itself. By combining a rigorously contractive mapping with Bayesian sampling, the method facilitates efficient posterior inference. Numerical experiments demonstrate that neglecting either parameter correlations or their uncertainties leads to substantial estimation bias, thereby validating the necessity and efficacy of treating unknown parameters as random variables with explicitly modeled dependence structures.
📝 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.