Variational Bayesian inference for comparison Var(1) models

Abstract

Suppose that we wish to determine which models in a candidate set are most likely to have given rise to a set of observed data. Then, it is well-established that, from a Bayesian viewpoint, evaluation of the marginal likelihood for each candidate is a crucial step to this end. For the purposes of model comparison, this will enable subsequent computation of both Bayes’ factors and posterior model probabilities. Given its evident significance in this area, it is thus regrettable that analytic calculation of the marginal likelihood is often not possible. To tackle this problem, one recent addition to the literature is the variational Bayesian approach. In this thesis, it is seen that variational Bayes provides efficient, accurate approximations to both the marginal likelihood and the parameter posterior distribution, conditioned on each model. In particular, the theory is applied to ranking sparse, vector autoregressive graphical models of order 1 in both the zero and non-zero mean case. That is, our primary aim is to estimate the unknown sparsity structure of the autoregressive matrix in the process. Moreover, approximate, marginal posterior information about the coefficients of this matrix is also of interest. To enable rapid exploration of higher-dimensional graphical spaces, a Metropolis-Hastings algorithm is presented so that a random walk can be made between neighbouring graphs. The scheme is then tested on both simulated and real datasets of varying dimension.