We develop an iterative framework for Bayesian inference problems where the posterior distribution may involve computationally intensive models, intractable gradients, significant posterior concentration, and pronounced non-Gaussianity. Our approach integrates: (i) a generalized annealing scheme that combines geometric tempering with multi-fidelity modeling; (ii) expressive measure transport surrogates for the intermediate annealed and final target distributions, learned variationally without evaluating gradients of the target density; and, (iii) an importance-weighting scheme to combine multiple quadrature rules, which recycles and reweighs expensive model evaluations as successive posterior approximations are built. Our scheme produces both a quadrature rule for computing posterior expectations and a transport-based approximation of the posterior from which we can easily generate independent Monte Carlo samples. We demonstrate the efficiency and accuracy of our approach on low-dimensional but strongly non-Gaussian Bayesian inverse problems involving partial differential equations.

翻译：我们针对后验分布可能涉及计算密集型模型、无法解析梯度、显著后验集中性及强非高斯性等贝叶斯推断问题，提出了一种迭代求解框架。该方法整合了：(i) 将几何退火与多保真建模相结合的广义退火方案；(ii) 基于度量传输的中间退火分布与最终目标分布的高表达性替代模型，通过变分法学习且无需计算目标密度的梯度；(iii) 基于重要性加权的多正交规则融合策略，在逐次构建后验逼近时，对昂贵的模型评估结果进行回收与重新赋权。该方案既可生成用于计算后验期望的正交规则，又能提供可便捷生成独立蒙特卡洛样本的传输型后验近似。我们通过涉及偏微分方程的低维强非高斯贝叶斯反问题，验证了该方法的效率与精度。