Academic

Bayesian inference is a very popular and effective methodology for statistical modeling. One common problem that arises in Bayesian inference is to calculate the expectation of some real-valued functions with respect to the posterior distribution. In such a scenario, sequential Monte Carlo (SMC) methods are powerful tools to sample sequentially from a sequence of probability distributions. In this research, we consider Bayesian inference for chain graph models and Bayesian inverse problems using SMC methods.

First, we consider Bayesian inference for partially observed Andersson-Madigan-Perlman (AMP) Gaussian chain graph models. Due to their ability to represent symmetric and non-symmetric relationships between the random variables of interest, such models are of particular interest in applications such as biological networks and financial time series. We develop a new Bayesian model for latent AMP chain graphs and introduce an SMC method as that improves upon Markov chain Monte Carlo (MCMC) based methods. Our approach is first illustrated on simulated data, and then applied to real case studies from university graduation rates and a pharmacokinetics study. We find the performance of our algorithm to be stable and robust.

Second, we consider the problem of estimating a parameter associated with a Bayesian inverse problem (BIP). The BIP is to infer unknown parameters associated with the solution of partial differential equations, and is often ill-posed and challenging to solve. We develop a new methodology to estimate the gradient of the log-likelihood w.r.t. the unknown parameter with no discretization bias. The proposed method provides an estimator which is not only unbiased, but also of finite variance. We show that the cost to achieve a given variance is very similar to the multilevel sequential Monte Carlo (MLSMC) approach. This is confirmed in numerical simulations. We further numerically investigate the performance of our estimator in the context of stochastic gradient algorithms.