Bayesian Learning in Undirected Graphical Models: Approximate MCMC Algorithms

Bayesian learning in undirected graphical models—computing posterior distributions over parameters and predictive quantities— is exceptionally difficult. We conjecture that for general undirected models, there are no tractable MCMC (Markov Chain Monte Carlo) schemes giving the correct equilibrium distribution over parameters. While this intractability, due to the partition function, is familiar to those performing parameter optimisation, Bayesian learning of posterior distributions over undirected model parameters has been unexplored and poses novel challenges. We propose several approximate MCMC schemes and test on fully observed binary models (Boltzmann machines) for a small coronary heart disease data set and larger artificial systems. While approximations must perform well on the model, their interaction with the sampling scheme is also important. Samplers based on variational mean-field approximations generally performed poorly, more advanced methods using loopy propagation, brief sampling and stochastic dynamics lead to acceptable parameter posteriors. Finally, we demonstrate these techniques on a Markov random field with hidden variables.