# Approximating Posterior Distributions in Belief Networks using Mixtures

Christopher M. Bishop, Microsoft Research, Cambridge
Neil D. Lawrence, University of Sheffield
Tommi S. Jaakkola, MIT
Michael I. Jordan, UC Berkeley

in Advances in Neural Information Processing Systems 10, pp 416-422

#### Abstract

Exact inference in densely connected Bayesian networks is computationally intractable, and so there is considerable interest in developing effective approximation schemes. One approach which has been adopted is to bound the log likelihood using a mean-field approximating distribution. While this leads to a tractable algorithm, the mean field distribution is assumed to be factorial and hence unimodal. In this paper we demonstrate the feasibility of using a richer class of approximating distributions based on mixtures of mean field distributions. We derive an efficient algorithm for updating the mixture parameters and apply it to the problem of learning in sigmoid belief networks. Our results demonstrate a systematic improvement over simple mean field theory as the number of mixture components is increased.

  @InProceedings{bishop-mixtures97, title = {Approximating Posterior Distributions in Belief Networks using Mixtures}, author = {Christopher M. Bishop and Neil D. Lawrence and Tommi S. Jaakkola and Michael I. Jordan}, booktitle = {Advances in Neural Information Processing Systems}, pages = {416}, year = {1998}, editor = {Michael I. Jordan and Michael J. Kearns and Sara A. Solla}, volume = {10}, address = {Cambridge, MA}, month = {00}, publisher = {MIT Press}, edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/1998-01-01-bishop-mixtures97.md}, url = {http://inverseprobability.com/publications/bishop-mixtures97.html}, abstract = {Exact inference in densely connected Bayesian networks is computationally intractable, and so there is considerable interest in developing effective approximation schemes. One approach which has been adopted is to bound the log likelihood using a mean-field approximating distribution. While this leads to a tractable algorithm, the mean field distribution is assumed to be factorial and hence unimodal. In this paper we demonstrate the feasibility of using a richer class of approximating distributions based on *mixtures* of mean field distributions. We derive an efficient algorithm for updating the mixture parameters and apply it to the problem of learning in sigmoid belief networks. Our results demonstrate a systematic improvement over simple mean field theory as the number of mixture components is increased.}, crossref = {Jordan:nips97}, key = {Bishop:mixtures97}, linkpsgz = {http://www.thelawrences.net/neil/mixtures.ps.gz}, OPTgroup = {} }
 %T Approximating Posterior Distributions in Belief Networks using Mixtures %A Christopher M. Bishop and Neil D. Lawrence and Tommi S. Jaakkola and Michael I. Jordan %B %C Advances in Neural Information Processing Systems %D %E Michael I. Jordan and Michael J. Kearns and Sara A. Solla %F bishop-mixtures97 %I MIT Press %P 416--422 %R %U http://inverseprobability.com/publications/bishop-mixtures97.html %V 10 %X Exact inference in densely connected Bayesian networks is computationally intractable, and so there is considerable interest in developing effective approximation schemes. One approach which has been adopted is to bound the log likelihood using a mean-field approximating distribution. While this leads to a tractable algorithm, the mean field distribution is assumed to be factorial and hence unimodal. In this paper we demonstrate the feasibility of using a richer class of approximating distributions based on *mixtures* of mean field distributions. We derive an efficient algorithm for updating the mixture parameters and apply it to the problem of learning in sigmoid belief networks. Our results demonstrate a systematic improvement over simple mean field theory as the number of mixture components is increased. 
 TY - CPAPER TI - Approximating Posterior Distributions in Belief Networks using Mixtures AU - Christopher M. Bishop AU - Neil D. Lawrence AU - Tommi S. Jaakkola AU - Michael I. Jordan BT - Advances in Neural Information Processing Systems PY - 1998/01/01 DA - 1998/01/01 ED - Michael I. Jordan ED - Michael J. Kearns ED - Sara A. Solla ID - bishop-mixtures97 PB - MIT Press SP - 416 EP - 422 UR - http://inverseprobability.com/publications/bishop-mixtures97.html AB - Exact inference in densely connected Bayesian networks is computationally intractable, and so there is considerable interest in developing effective approximation schemes. One approach which has been adopted is to bound the log likelihood using a mean-field approximating distribution. While this leads to a tractable algorithm, the mean field distribution is assumed to be factorial and hence unimodal. In this paper we demonstrate the feasibility of using a richer class of approximating distributions based on *mixtures* of mean field distributions. We derive an efficient algorithm for updating the mixture parameters and apply it to the problem of learning in sigmoid belief networks. Our results demonstrate a systematic improvement over simple mean field theory as the number of mixture components is increased. ER - 
 Bishop, C.M., Lawrence, N.D., Jaakkola, T.S. & Jordan, M.I.. (1998). Approximating Posterior Distributions in Belief Networks using Mixtures. Advances in Neural Information Processing Systems 10:416-422