Variational inference in nonconjugate models

Mean-field variational methods are widely used for approximate posterior inference in many prob-abilistic models. In a typical application, mean-field methods approximately compute the posterior with a coordinate-ascent optimization algorithm. When the model is conditionally conjugate, the coordinate updates are easily derived and in closed form. However, many models of interest—like the correlated topic model and Bayesian logistic regression—are nonconjugate. In these models, mean-field methods cannot be directly applied and practitioners have had to develop variational algorithms on a case-by-case basis. In this paper, we develop two generic methods for nonconju-gate models, Laplace variational inference and delta method variational inference. Our methods have several advantages: they allow for easily derived variational algorithms with a wide class of nonconjugate models; they extend and unify some of the existing algorithms that have been derived for specific models; and they work well on real-world data sets. We studied our methods on the correlated topic model, Bayesian logistic regression, and hierarchical Bayesian logistic regression.