Second-Order Latent-Space Variational Bayes for Approximate Bayesian Inference

In this letter, we consider a variational approximate Bayesian inference framework, latent-space variational Bayes (LSVB), in the general context of conjugate-exponential family models with latent variables. In the LSVB approach, we integrate out model parameters in an exact way and then perform the variational inference over only the latent variables. It can be shown that LSVB can achieve better estimates of the model evidence as well as the distribution over the latent variables than the popular variational Bayesian expectation-maximization (VBEM). However, the distribution over the latent variables in LSVB has to be approximated in practice. As an approximate implementation of LSVB, we propose a second-order LSVB (SoLSVB) method. In particular, VBEM can be derived as a special case of a first-order approximation in LSVB (Sung). SoLSVB can capture higher order statistics neglected in VBEM and can therefore achieve a better approximation. Examples of Gaussian mixture models are used to illustrate the comparison between our method and VBEM, demonstrating the improvement.