这篇论文给神经似然逼近奠定了凸优化理论基础，证明训练目标严格凸且样本量增大时收敛，适合数学基础扎实的研究者了解新方法。
该论文针对高维贝叶斯反问题中传统MCMC计算成本高的问题，提出通过最小化真实后验与近似后验之间的KL散度来训练神经似然代理。作者证明，使用未归一化势并将归一化纳入训练目标后，学习问题变为严格凸优化。理论表明，随着样本量增大，数据驱动目标的经验最小化器收敛到真实似然。数值实验在去模糊和非线性PDE成像问题上验证了该方法。
A Convex Approximation Framework for Neural Likelihood-Based Bayesian Inverse Problems
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.