Disentangled Deep Priors for Bayesian Inverse Problems

2026-04-02

AI Summary

This work addresses the challenges of interpretability and robustness in high-dimensional Bayesian inverse problems by proposing a disentangled deep generative prior. The latent space is explicitly partitioned into interpretable variables aligned with known physical parameters and residual variables capturing remaining uncertainties. By solving the inverse problem in the latent space using a linearized generator—combined with MAP estimation, MCMC sampling, and hierarchical Bayesian inference—the method introduces representation disentanglement into Bayesian prior design for the first time. Evaluated on elliptic PDE inverse problems, the approach matches the performance of an oracle Gaussian process prior under correct model specification and substantially outperforms it under model misspecification, while accurately recovering physical parameters and producing spatially 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.

Problem

• Bayesian inverse problems
• disentangled representation
• deep generative models
• uncertainty quantification
• interpretable parameters

Innovation

• disentangled deep generative model
• Bayesian inverse problems
• structured prior
• latent-space inference
• uncertainty quantification