A Convex Approximation Framework for Neural Likelihood-Based Bayesian Inverse Problems

• URL: http://arxiv.org/abs/2607.06252v1
• Date: Tue, 07 Jul 2026 13:18:45 GMT
• Title: A Convex Approximation Framework for Neural Likelihood-Based Bayesian Inverse Problems
• Abstract summary: We study neural likelihood approximation, where the goal is to learn the likelihood function directly from data without explicit knowledge of the underlying data-generating process.<n>We show that empirical minimizers of the resulting data-driven objective converge to the true likelihood as the sample size grows.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Many problems in science and engineering are difficult to model accurately, either due to unknown physical mechanisms, poorly quantified measurement uncertainty, or prohibitive computational costs of high-fidelity simulations. These challenges limit the applicability of classical probabilistic inference methods such as Markov chain Monte Carlo, especially in high-dimensional Bayesian inverse problems. As data from scientific experiments become increasingly available, machine learning methods offer a flexible alternative to explicit parametric modelling. We study neural likelihood approximation, where the goal is to learn the likelihood function directly from data without explicit knowledge of the underlying data-generating process. A common approach trains likelihood surrogates by minimizing the Kullback-Leibler divergence between the true posterior and an approximate posterior, which is equivalent to minimizing the expected negative log-likelihood. This work improves the theoretical foundations of neural likelihood approximation by alleviating limitations of restrictive model classes: we show that, by working with un-normalized potentials and folding normalization into the training objective, the resulting learning problem is strictly convex. We show that empirical minimizers of the resulting data-driven objective converge to the true likelihood as the sample size grows. Numerical experiments for the neural likelihood approximation are conducted for a deblurring and a non-linear PDE based imaging problem.

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