Bayesian learning of forest and tree graphical models

Frequentist methods for learning Gaussian graphical model structure are unsuccessful at identifying hubs when n < p. An alternative is Bayesian structure-learning, in which it is common to restrict attention to certain classes of graphs and to explore and approximate the posterior distribution by repeatedly moving from one graph to another, using MCMC or other methods such as stochastic shotgun search (SSS). ( give two corrected versions of an algorithm for non-decomposable graphs and discuss random graph distributions in depth, in particular as priors in Bayesian structure-learning. The main topic of the thesis is Bayesian structure-learning with forests or trees. Forest and tree graphical models are widely used, and I explain how restricting attention to these graphs can be justified using theorems on random graphs. I describe how to use methods based on the Chow-Liu algorithm and the Matrix Tree Theorem to find the MAP forest and certain quantities in the full posterior distribution on trees. I give adapted versions of MCMC and SSS for approximating the posterior distribution for forests and trees, and systems for storing these graphs so that it is easy and efficient to choose legal moves to neighbouring forests or trees and update the stored information. Experiments with the adapted algorithms and simulated datasets show that the system for storing trees so that moves are chosen uniformly at random does not bring much advantage over simpler systems. SSS with trees does well when the true graph is a tree or a sparse graph. Graph priors improve detection of hubs but need large ranges of probabilities to have much effect. SSS with trees and SSS with forests do better than SSS with decomposable graphs in certain cases. MCMC on forests often fails to mix well and MCMC on trees is much slower than SSS.EThOS - Electronic Theses Online ServiceGBUnited Kingdo