Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators

This paper concerns the use of the expectation-maximisation (EM) algorithm for inference in partially observed diffusion processes. In this context, a well known problem is that all except a few diffusion processes lack closed-form expressions of the transition densities. Thus, in order to estimate efficiently the EM intermediate quantity we construct, using novel techniques for unbiased estimation of diffusion transition densities, a random weight fixed-lag auxiliary particle smoother, which avoids the well known problem of particle trajectory degeneracy in the smoothing mode. The estimator is justified theoretically and demonstrated on a simulated example

Nonlinear Filtering for Stochastic Volatility Models with Heavy Tails and Leverage

This paper develops a computationally efficient filtering based procedure for the estimation of the heavy tailed SV model with leverage. While there are many accepted techniques for the estimation of standard SV models, incorporating these effects into an SV framework is difficult. Simulation evidence provided in this paper indicates that the proposed procedure outperforms competing approaches in terms of the accuracy of parameter estimation. In an empirical setting, it is shown how the individual effects of heavy tails and leverage can be isolated using standard likelihood ratio tests.

Variational Approximate Inference in Latent Linear Models

Latent linear models are core to much of machine learning and statistics. Specific examples of this model class include Bayesian generalised linear models, Gaussian process regression models and unsupervised latent linear models such as factor analysis and principal components analysis. In general, exact inference in this model class is computationally and analytically intractable. Approximations are thus required. In this thesis we consider deterministic approximate inference methods based on minimising the Kullback-Leibler (KL) divergence between a given target density and an approximating `variational' density. First we consider Gaussian KL (G-KL) approximate inference methods where the approximating variational density is a multivariate Gaussian. Regarding this procedure we make a number of novel contributions: sufficient conditions for which the G-KL objective is differentiable and convex are described, constrained parameterisations of Gaussian covariance that make G-KL methods fast and scalable are presented, the G-KL lower-bound to the target density's normalisation constant is proven to dominate those provided by local variational bounding methods. We also discuss complexity and model applicability issues of G-KL and other Gaussian approximate inference methods. To numerically validate our approach we present results comparing the performance of G-KL and other deterministic Gaussian approximate inference methods across a range of latent linear model inference problems. Second we present a new method to perform KL variational inference for a broad class of approximating variational densities. Specifically, we construct the variational density as an affine transformation of independently distributed latent random variables. The method we develop extends the known class of tractable variational approximations for which the KL divergence can be computed and optimised and enables more accurate approximations of non-Gaussian target densities to be obtained