Sampling through iterated approximation: Gradient-free and multi-fidelity Bayesian inference via transport
This addresses computational challenges in Bayesian inference for researchers in fields like inverse problems, but it is incremental as it builds on existing transport and annealing methods.
The paper tackles Bayesian inference with computationally intensive models and non-Gaussian posteriors by developing an iterative framework that integrates annealing, transport surrogates, and importance weighting, achieving efficient and accurate results on low-dimensional inverse problems.
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.