Computational Mathematics and Scientific Computing Seminar

Many practical Bayesian inference problems fall into the "likelihood-free" setting, where evaluations of the likelihood function are unavailable or intractable. I will discuss how transportation of measure can solve such problems, by pushing prior samples, or samples from a joint parameter-data prior, to the desired conditional distribution. These methods have broad utility for inference in stochastic/generative models, as well as for data assimilation problems motivated by geophysical applications. As an example of the latter, I will present a new approach to nonlinear ensemble filtering that uses (sparse) triangular transport maps to produce robust approximations of the filtering distribution in high dimensions. The approach can be understood as the natural generalization of the ensemble Kalman filter (EnKF) to nonlinear updates, and can reduce the intrinsic bias of the EnKF at a marginal increase in computational cost.

 

Underlying the analysis step of this filtering scheme, and the transport approach to likelihood-free inference in general, is the need to construct monotone triangular maps from relatively small samples. I will describe a systematic framework for representing and learning such maps, via invertible transformations of smooth functions, and demonstrate that the associated minimization problem has many favorable properties. In particular, we devise a sample-efficient adaptive algorithm that estimates sparse approximations of triangular transport maps. I will also contrast different ways of extracting prior-to-posterior transformations in likelihood-free inference, highlighting a composition-of-maps approach that significantly improves performance.

 

This is joint work with Ricardo Baptista, Alessio Spantini, and Olivier Zahm.