Machine Learning

, Volume 37, Issue 2, pp 183-233

First online:

An Introduction to Variational Methods for Graphical Models

  • Michael I. JordanAffiliated withDepartment of Electrical Engineering and Computer Sciences and Department of Statistics, University of California
  • , Zoubin GhahramaniAffiliated withGatsby Computational Neuroscience Unit, University College
  • , Tommi S. JaakkolaAffiliated withArtificial Intelligence Laboratory, MIT
  • , Lawrence K. SaulAffiliated withAT&T Labs–Research


This paper presents a tutorial introduction to the use of variational methods for inference and learning in graphical models (Bayesian networks and Markov random fields). We present a number of examples of graphical models, including the QMR-DT database, the sigmoid belief network, the Boltzmann machine, and several variants of hidden Markov models, in which it is infeasible to run exact inference algorithms. We then introduce variational methods, which exploit laws of large numbers to transform the original graphical model into a simplified graphical model in which inference is efficient. Inference in the simpified model provides bounds on probabilities of interest in the original model. We describe a general framework for generating variational transformations based on convex duality. Finally we return to the examples and demonstrate how variational algorithms can be formulated in each case.

graphical models Bayesian networks belief networks probabilistic inference approximate inference variational methods mean field methods hidden Markov models Boltzmann machines neural networks