Recursive inference for inverse problems using variational Bayes methodology

Abstract

The epistemic inverse problem is treated via Bayesian inference in this paper. In particular, the conditions for recursive computation of the required posterior inference are recalled. The emphasis is on appropriate structure in the observation model and choice by the Bayesian designer of a prior matched to that structure. Bayesian conjugate inference for the exponential family (EF) of observation models is recalled. This inspires progress with design of recursive algorithms for the time-variant (Bayesian filtering) case, using the variational Bayes (VB) approximation. A rich class of augmented observation models is defined, for which the posterior inference is closed under a local VB approximation, a principle known as VB-conjugacy. The key mathematical object is the VB-observation model, arising from application of VB in each data step. We force this to be an EF member. The theory is specialized to finite mixtures of heterogenous components, requiring recursive evaluation of one sufficient statistic per component, with the posterior component weights (i.e. filtering distribution) evaluated in a principled way. Further specialization to signal-independent system modelling is also considered. An extended case study in decoding and synchronization for the phase-uncertain digital receiver is presented as a key application of VB-conjugate recursive inference.