Truncated Variational Expectation Maximization

We derive a novel variational expectation maximization approach based on truncated posterior distributions. Truncated distributions are proportional to exact posteriors within subsets of a discrete state space and equal zero otherwise. The treatment of the distributions' subsets as variational parameters distinguishes the approach from previous variational approaches. The specific structure of truncated distributions allows for deriving novel and mathematically grounded results, which in turn can be used to formulate novel efficient algorithms to optimize the parameters of probabilistic generative models. Most centrally, we find the variational lower bounds that correspond to truncated distributions to be given by very concise and efficiently computable expressions, while update equations for model parameters remain in their standard form. Based on these findings, we show how efficient and easily applicable meta-algorithms can be formulated that guarantee a monotonic increase of the variational bound. Example applications of the here derived framework provide novel theoretical results and learning procedures for latent variable models as well as mixture models. Furthermore, we show that truncated variation EM naturally interpolates between standard EM with full posteriors and EM based on the maximum a-posteriori state (MAP). The approach can, therefore, be regarded as a generalization of the popular `hard EM' approach towards a similarly efficient method which can capture more of the true posterior structure.