Nadège Polette | Searcharxiv

arXiv2026

Covariance-Informed Subspace: an Adaptive Gradient-Free Input Dimension Reduction Method for Bayesian Inference

This paper addresses the challenge of dimension reduction (DR) in Bayesian inference of high-resolution two-or three-dimensional fields, where a priori parametrizations require a large number of terms. The underlying idea is common to state-of-the-art methods in which the parameter space is decomposed into two subspaces, one informed by the likelihood and one constrained by the prior. DR techniques generally use gradient information from the log-likelihood to derive the corresponding subspaces. However, the gradient may be unavailable or expensive to compute accurately, for instance in the case of simulation-based inference. Inspired by approaches based on likelihood-informed subspaces, we develop a new DR method tailored for settings where gradient computation is not feasible. More specifically, we propose a gradient-free indicator for determining whether a direction is informed by the data. This indicator is derived from the posterior-to-prior covariance ratio introduced in Spantini et al. (2015). We show that, in the linear Gaussian case, this indicator combined with an approximate likelihood leads to a better posterior approximation. The method is then extended to nonlinear cases, and strategies to approximate the posterior covariance are detailed. We demonstrate the effectiveness of this DR through two high-dimensional inference problems arising from groundwater and atmospheric applications.

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arXiv2025

Change of Measure for Bayesian Field Inversion with Hierarchical Hyperparameters Sampling

This paper proposes an effective treatment of hyperparameters in the Bayesian inference of a scalar field from indirect observations. Obtaining the joint posterior distribution of the field and its hyperparameters is challenging. The infinite dimensionality of the field requires a finite parametrization that usually involves hyperparameters to reflect the limited prior knowledge. In the present work, we consider a Karhunen-Lo{è}ve(KL) decomposition for the random field and hyperparameters to account for the lack of prior knowledge of its autocovariance function. The hyperparameters must be inferred. To efficiently sample jointly the KL coordinates of the field and the autocovariance hyperparameters, we introduce a change of measure to reformulate the joint posterior distribution into a hierarchical Bayesian form. The likelihood depends only onthe field's coordinates in a fixed KL basis, with a prior conditioned on the hyperparameters. We exploit this structure to derive an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial Chaos expansions) of the forward model predictions to further accelerate the MCMC sampling. A first application to a transient diffusionproblem shows that our method is consistent with other approaches based on a change of coordinates (Sraj et al., 2016). A second application to a seismic traveltime tomography highlights the importance of inferring the hyperparameters. A third application to a 2D anisotropic groundwater flow problem illustrates the method on a more complex geometry.

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