Disentangled Deep Priors for Bayesian Inverse Problems

TL;DR

This paper introduces a structured disentangled deep prior for Bayesian inverse problems, enabling interpretable uncertainty quantification and improved robustness in high-dimensional settings.

Contribution

It develops a hierarchical prior using a disentangled deep generative model, linking representation-level disentanglement to uncertainty separation in inverse problems.

Findings

Matches oracle Gaussian process prior under correct specification.

Provides substantial improvement under prior misspecification.

Recovers interpretable physical parameters with calibrated uncertainty estimates.

Abstract

We propose a structured prior for high-dimensional Bayesian inverse problems based on a disentangled deep generative model whose latent space is partitioned into auxiliary variables aligned with known and interpretable physical parameters and residual variables capturing remaining unknown variability. This yields a hierarchical prior in which interpretable coordinates carry domain-relevant uncertainty while the residual coordinates retain the flexibility of deep generative models. By linearizing the generator, we characterize the induced prior covariance and derive conditions under which the posterior exhibits approximate block-diagonal structure in the latent variables, clarifying when representation-level disentanglement translates into a separation of uncertainty in the inverse problem. We formulate the resulting latent-space inverse problem and solve it using MAP estimation and Markov chain Monte Carlo (MCMC) sampling. On elliptic PDE inverse problems, such as conductivity identification and source identification, the approach matches an oracle Gaussian process prior under correct specification and provides substantial improvement under prior misspecification, while recovering interpretable physical parameters and producing spatially calibrated uncertainty estimates.