Many practical Bayesian inference problems fall into the "likelihood-free" setting, where evaluations of the likelihood function or prior density are unavailable or intractable; instead one can only simulate (i.e., draw samples from) the associated distributions. I will discuss how transportation of measure can help solve such problems, by constructing maps that push prior samples, or samples from a joint parameter-data prior, to the desired conditional distribution. These methods have broad utility for inference in stochastic and generative models, as well as for data assimilation problems motivated by geophysical applications. Key issues in this construction center on: (1) the estimation of transport maps from few samples; and (2) parameterizations of monotone maps. I will discuss developments on both fronts, including some recent efforts in joint dimension reduction for conditional sampling.
As an example, I will present an approach to nonlinear filtering in dynamical systems which 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.
This is joint work with Ricardo Baptista, Alessio Spantini, Olivier Zahm, and Jakob Zech.
Youssef Marzouk will join our SFB Kick-off meeting online from the US. We will meet in house 27 to listen to the talk collectively.