Correcting the Laplace Method with Variational Bayes

Bayesian methods involve a prior belief about a model and learning from the data to arrive at a new belief, which is termed the posterior belief. Mathematically, the posterior belief can be derived from the prior belief and the empirical evidence presented by the data using Bayes' rule. In this way Bayesian analysis is a natural statistical machine learning method (see [42, 9, 33, 34, 40, 46, 30, 35] amongst many others), and especially powerful for small datasets, missing data or complex models. From a computational viewpoint, various approaches have been proposed to perform Bayesian analysis, mainly exact (analytical or sampling-based) or approximate inferential approaches. Sampling-based methods like Markov Chain Monte Carlo (MCMC) sampling with its extensions (see [28, 12, 8, 1], amongst others) gained popularity in the 1990's but suffers from slow speed and convergence issues exacerbated by large data and/or complicated models.