Boltzmann machines are undirected graphical models with two-state stochastic variables, in which the logarithms of the clique potentials are quadratic functions of the node states. They have been widely studied in the neural computing literature, although their practical applicability has been limited by the difficulty of finding an effective learning algorithm. One well-established approach, known as mean field theory, represents the stochastic distribution using a factorized approximation. However, the corresponding learning algorithm often fails to find a good solution. We conjecture that this is due to the implicit uni-modality of the mean field approximation which is therefore unable to capture multi-modality in the true distribution. In this paper we use variational methods to approximate the stochastic distribution using multi-modal mixtures of factorized distributions. We present results for both inference and learning to demonstrate the effectiveness of this approach.

@InProceedings{lawrence-mixtures98,
title = {Mixture Representations for Inference and Learning in Boltzmann Machines},
author = {Neil D. Lawrence and Christopher M. Bishop and Michael I. Jordan},
booktitle = {Uncertainty in Artificial Intelligence},
pages = {320},
year = {1998},
editor = {Gregory F. Cooper and Serafín Moral},
volume = {14},
address = {San Francisco, CA},
month = {00},
publisher = {Morgan Kauffman},
edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/1998-01-01-lawrence-mixtures98.md},
url = {http://inverseprobability.com/publications/lawrence-mixtures98.html},
abstract = {Boltzmann machines are undirected graphical models with two-state stochastic variables, in which the logarithms of the clique potentials are quadratic functions of the node states. They have been widely studied in the neural computing literature, although their practical applicability has been limited by the difficulty of finding an effective learning algorithm. One well-established approach, known as mean field theory, represents the stochastic distribution using a factorized approximation. However, the corresponding learning algorithm often fails to find a good solution. We conjecture that this is due to the implicit uni-modality of the mean field approximation which is therefore unable to capture multi-modality in the true distribution. In this paper we use variational methods to approximate the stochastic distribution using multi-modal *mixtures* of factorized distributions. We present results for both inference and learning to demonstrate the effectiveness of this approach.},
crossref = {Cooper:uai98},
key = {Lawrence:mixtures98},
linkpsgz = {http://www.thelawrences.net/neil/boltzmann.ps.gz},
OPTgroup = {}
}

%T Mixture Representations for Inference and Learning in Boltzmann Machines
%A Neil D. Lawrence and Christopher M. Bishop and Michael I. Jordan
%B
%C Uncertainty in Artificial Intelligence
%D
%E Gregory F. Cooper and Serafín Moral
%F lawrence-mixtures98
%I Morgan Kauffman
%P 320--327
%R
%U http://inverseprobability.com/publications/lawrence-mixtures98.html
%V 14
%X Boltzmann machines are undirected graphical models with two-state stochastic variables, in which the logarithms of the clique potentials are quadratic functions of the node states. They have been widely studied in the neural computing literature, although their practical applicability has been limited by the difficulty of finding an effective learning algorithm. One well-established approach, known as mean field theory, represents the stochastic distribution using a factorized approximation. However, the corresponding learning algorithm often fails to find a good solution. We conjecture that this is due to the implicit uni-modality of the mean field approximation which is therefore unable to capture multi-modality in the true distribution. In this paper we use variational methods to approximate the stochastic distribution using multi-modal *mixtures* of factorized distributions. We present results for both inference and learning to demonstrate the effectiveness of this approach.

TY - CPAPER
TI - Mixture Representations for Inference and Learning in Boltzmann Machines
AU - Neil D. Lawrence
AU - Christopher M. Bishop
AU - Michael I. Jordan
BT - Uncertainty in Artificial Intelligence
PY - 1998/01/01
DA - 1998/01/01
ED - Gregory F. Cooper
ED - Serafín Moral
ID - lawrence-mixtures98
PB - Morgan Kauffman
SP - 320
EP - 327
UR - http://inverseprobability.com/publications/lawrence-mixtures98.html
AB - Boltzmann machines are undirected graphical models with two-state stochastic variables, in which the logarithms of the clique potentials are quadratic functions of the node states. They have been widely studied in the neural computing literature, although their practical applicability has been limited by the difficulty of finding an effective learning algorithm. One well-established approach, known as mean field theory, represents the stochastic distribution using a factorized approximation. However, the corresponding learning algorithm often fails to find a good solution. We conjecture that this is due to the implicit uni-modality of the mean field approximation which is therefore unable to capture multi-modality in the true distribution. In this paper we use variational methods to approximate the stochastic distribution using multi-modal *mixtures* of factorized distributions. We present results for both inference and learning to demonstrate the effectiveness of this approach.
ER -

Lawrence, N.D., Bishop, C.M. & Jordan, M.I.. (1998). Mixture Representations for Inference and Learning in Boltzmann Machines. Uncertainty in Artificial Intelligence 14:320-327