# Markovian inference in belief networks

Brendan J. Frey
Neil D. Lawrence, University of Sheffield
Christopher M. Bishop, Microsoft Research, Cambridge

#### Abstract

Bayesian belief networks can represent the complicated probabilistic processes that form natural sensory inputs. Once the parameters of the network have been learned,nonlinear inferences about the input can be made by computing the posterior distribution over the hidden units (e.g., depth in stereo vision) given the input. Computing the posterior distribution exactly is not practical in richly-connected networks, but it turns out that by using a variational (a.k.a., mean field) method, it is easy to find a product-form distribution that approximates the true posterior distribution. This approximation assumes that the hidden variables are independent given the current input. In this paper, we explore a more powerful variational technique that models the posterior distribution using a Markov chain. We compare this method with inference using mean fields and mixtures of mean fields in randomly generated networks.

  @TechReport{frey-markovian98, title = {Markovian inference in belief networks}, author = {Brendan J. Frey and Neil D. Lawrence and Christopher M. Bishop}, year = {1998}, institution = {The Beckman Institute}, address = {University of Illinois at Urbana-Champaign, 405 North Mathews Avenue, Urbana, IL 61801, USA}, month = {00}, edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/1998-01-01-frey-markovian98.md}, url = {http://inverseprobability.com/publications/frey-markovian98.html}, abstract = {Bayesian belief networks can represent the complicated probabilistic processes that form natural sensory inputs. Once the parameters of the network have been learned,nonlinear inferences about the input can be made by computing the posterior distribution over the hidden units (e.g., depth in stereo vision) given the input. Computing the posterior distribution exactly is not practical in richly-connected networks, but it turns out that by using a variational (a.k.a., mean field) method, it is easy to find a product-form distribution that approximates the true posterior distribution. This approximation assumes that the hidden variables are independent given the current input. In this paper, we explore a more powerful variational technique that models the posterior distribution using a Markov chain. We compare this method with inference using mean fields and mixtures of mean fields in randomly generated networks.}, key = {Frey:Markovian98}, note = {Originally submitted to \emph{NIPS 1998}}, linkpsgz = {http://www.thelawrences.net/neil/mi.ps.gz}, OPTgroup = {} }
 %T Markovian inference in belief networks %A Brendan J. Frey and Neil D. Lawrence and Christopher M. Bishop %B %D %F frey-markovian98 %P -- %R %U http://inverseprobability.com/publications/frey-markovian98.html %X Bayesian belief networks can represent the complicated probabilistic processes that form natural sensory inputs. Once the parameters of the network have been learned,nonlinear inferences about the input can be made by computing the posterior distribution over the hidden units (e.g., depth in stereo vision) given the input. Computing the posterior distribution exactly is not practical in richly-connected networks, but it turns out that by using a variational (a.k.a., mean field) method, it is easy to find a product-form distribution that approximates the true posterior distribution. This approximation assumes that the hidden variables are independent given the current input. In this paper, we explore a more powerful variational technique that models the posterior distribution using a Markov chain. We compare this method with inference using mean fields and mixtures of mean fields in randomly generated networks. 
 TY - CPAPER TI - Markovian inference in belief networks AU - Brendan J. Frey AU - Neil D. Lawrence AU - Christopher M. Bishop PY - 1998/01/01 DA - 1998/01/01 ID - frey-markovian98 SP - EP - UR - http://inverseprobability.com/publications/frey-markovian98.html AB - Bayesian belief networks can represent the complicated probabilistic processes that form natural sensory inputs. Once the parameters of the network have been learned,nonlinear inferences about the input can be made by computing the posterior distribution over the hidden units (e.g., depth in stereo vision) given the input. Computing the posterior distribution exactly is not practical in richly-connected networks, but it turns out that by using a variational (a.k.a., mean field) method, it is easy to find a product-form distribution that approximates the true posterior distribution. This approximation assumes that the hidden variables are independent given the current input. In this paper, we explore a more powerful variational technique that models the posterior distribution using a Markov chain. We compare this method with inference using mean fields and mixtures of mean fields in randomly generated networks. ER - 
 Frey, B.J., Lawrence, N.D. & Bishop, C.M.. (1998). Markovian inference in belief networks.:-