On the Prior and Posterior Distributions Used in Graphical Modelling
This work addresses a methodological gap for researchers in graphical modeling, but it is incremental as it builds on existing Bayesian techniques without introducing a new paradigm.
The paper tackles the problem of insufficiently studied prior and posterior distributions over graph structures in graphical model learning, providing a characterization of their behavior as a function of possible edges and deriving measures of structural variability for Bayesian and Markov networks.
Graphical model learning and inference are often performed using Bayesian techniques. In particular, learning is usually performed in two separate steps. First, the graph structure is learned from the data; then the parameters of the model are estimated conditional on that graph structure. While the probability distributions involved in this second step have been studied in depth, the ones used in the first step have not been explored in as much detail. In this paper, we will study the prior and posterior distributions defined over the space of the graph structures for the purpose of learning the structure of a graphical model. In particular, we will provide a characterisation of the behaviour of those distributions as a function of the possible edges of the graph. We will then use the properties resulting from this characterisation to define measures of structural variability for both Bayesian and Markov networks, and we will point out some of their possible applications.