Efficient Importance Sampling Techniques for Large Dimensional and Multimodal Posterior Computations

In recent work, we proposed some new ideas for efficient sequential importance sampling in the context of particle filtering. These were specifically designed for problems with multimodal posteriors (particularly those with multimodal likelihoods) and with very large dimensions. In this work, we demonstrate the use of similar ideas to improve the performance of importance sampling (IS) in static problems. The key idea of our proposed method is to split the state space in such a way that the posterior conditioned on a small part of the state space is "unimodal". We can then importance sample from the prior for the small "multimodal" part of the state space while adapting existing efficient IS techniques for the much larger dimensional "unimodal" part. We give a modified version of a result from our recent work to obtain sufficient conditions to ensure posterior unimodality. Also, for a subspace of the "unimodal" state space having small enough prior variance, one can replace IS by just estimating the conditional posterior mode. We call this the mode tracking (MT) approximation of IS. We show, via experiments on a large dimensional temperature field estimation problem, that when the number of samples, N, is small, the MT approximation outperforms any standard IS technique.