Sampling through iterated approximation: Gradient-free and multi-fidelity Bayesian inference via transport

Abstract

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.

Figures (12)

• Figure 1: Illustration of a single step ${j}$ of our method, proceeding from top left to bottom right. The relationship between choosing $\beta_{{j}}$ and reweighting $\widetilde{Q}_{j}$ is bidirectional: we select parameter $\beta_{{j}}$ to ensure high-quality weights $w_{j}^{(k)}$.
• Figure 1: Likelihood ratios induced by the three PDE discretizations in \ref{['sec:diffusion_single']}.
• Figure 1: Output quadratures of MCMC (left) and SMC (right). SMC marker size and color reflect the weight and log-weight of the sample point, respectively; MCMC is shown with opacity to demonstrate the rejections in the Markov chain.
• Figure 2: Quadrature rules $Q_{j}^\mathrm{MIS}$ for \ref{['sec:diffusion_single']}. Both color and size reflect the weight of a point, with uniform weights desired.
• Figure 3: Visualizing the results for \ref{['sec:diffusion_single']}. Columns from left to right: target density ${\pi}_j$, transport-induced surrogate density $\widetilde{{\pi}}_{j}$, samples from the surrogate, and weighted log-ratio of ${\pi}_j$ to $\widetilde{{\pi}}_{j}$.