Variational inference for stochastic processes

Abstract

Stochastic process models such as stochastic differential equations (SDEs), state-space models (SSMs), Gaussian processes (GPs) and latent force models (LFMs), provide a powerful collection of modelling techniques to better our understanding of many physical systems. In treating these models within the Bayesian paradigm, we further yield a rich expression of our uncertainty, and gain the ability to incorporate our prior beliefs. However, performing Bayesian posterior inference is not without significant challenge. Exact likelihood calculations can often be intractable, take an infeasibly long time to compute, or be challenging to approximate in the presence of missing data. Therefore, designing new approaches to perform Bayesian inference for this family of stochastic process models is of great scientific interest. Variational inference (VI) has had great success is scaling Bayesian inference across a range of problem domains. Historically, however, its successful application to stochastic process models has been limited. The reason is two-fold. Firstly, mini-batch likelihood estimation techniques often employed by VI have only previously been applicable to models of independent data. Secondly, approximating distributions have often imposed unrealistic assumptions over the posterior. Fortunately, however, recent advances in generative modelling have provided the framework with which to solve these problems. Here, artificial neural networks can be used to flexibly construct powerful density approximations, which are then amenable to fast computation using modern GPUs. This is otherwise known as black-box-variational inference. This thesis presents a collection of black-box variational methods for the purposes of approximate inference in SDEs, SSMs, GPs and LFMs. Here we leverage artificial neural networks to parametrise our approximate posterior distributions, permitting accurate inference in a short time. We begin by presenting two methods for SDE inference. The first, inspired by the Euler-Maruyama discretisation, approximates the discrete-time solution to a conditioned diffusion process using recurrent neural networks. The second, which extends the first, eschews a discretisation scheme and approximates the continuous-time process directly. Finally we consider the use of normalising flows for inference using SSMs (including discrete-time SDEs), GPs and LFMs. Here we design a generative architecture that permits mini-batch optimization, allowing approximate inference for big data