Title: Variational Bayesian Methods for Inferring Spatial Statistics and Nonlinear Dynamics
Description: This thesis discusses four novel statistical methods and approximate inference techniques for analyzing structured neural and molecular sequence data. The main contributions are new algorithms for approximate inference and learning in Bayesian latent variable models involving spatial statistics and nonlinear dynamics. First, we propose an amortized variational inference method to separate a set of overlapping signals into spatially localized source functions without knowledge of the original signals or the mixing process. In the second part of this dissertation, we discuss two approaches for uncovering nonlinear, smooth latent dynamics from sequential data. Both algorithms construct variational families on extensions of nonlinear state space models where the underlying systems are described by hidden stochastic differential equations. The first method proposes a structured approximate posterior describing spatially-dependent linear dynamics, as well as an algorithm that relies on the fixed-point iteration method to achieve convergence.

The second method proposes a variational backward simulation technique from an unbiased estimate of the marginal likelihood defined through a subsampling process. In the final chapter, we develop connections between discrete and continuous variational sequential search for Bayesian phylogenetic inference. We propose a technique that uses sequential search to construct a variational objective defined on the composite space of non-clock phylogenetic trees. Each of these techniques are motivated by real problems within computational biology and applied to provide insights into the underlying structure of complex data.