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

What the math gives to ML

The paper's transferable asset is the replacement of a normalized likelihood model by an unnormalized potential whose partition function is included explicitly in the training objective. When the potential is linear in its trainable coefficients, the posterior-fitting objective is convex, and its Hessian is a covariance matrix under the model posterior; strict convexity follows when the features are identifiable. This suggests a practical hybrid architecture: use a neural encoder to generate fixed or slowly updated features, but train the final Bayesian potential head with a globally solvable convex objective. The main engineering challenge is estimating the latent-space partition function and posterior expectations, so the first implementation should use a low-dimensional latent space or importance/Monte Carlo samples.