M. Studeny: Mathematical aspects of learning Bayesian networks: Bayesian quality criteria. Research report n. 2234, Institute of Information Theory and Automation, Prague, December 2008.

The motivation for this research report is learning a Bayesian network (BN) structure by the method of maximizing a quality criterion. The aim is to summarize the mathematical grounding for the Bayesian approach to learning a BN structure. At first, some of basic statistical concepts are recapitulated. Then the classes of multinomial and Dirichlet distributions are dealt with in more detail. A peculiar question what is, in fact, the correct dominating measure for (the class of) Dirichlet distributions is answered. After that basic Bayesian terminology is recalled and the (statistical) model of a discrete BN is formally introduced. It is shown to be an exponential family. This allows one to introduce a Bayesian model for (learning discrete) BN structures, including explicit specification of the mathematical assumptions taken from the literature. This leads to the formula for the (data vector of the) corresponding Bayesian quality criterion (= the logarithm of the marginal likelihood).

AMS classification 68T30

learning Bayesian network structure
statistical model
multinomial distribution
Dirichlet distribution
exponential family
Bayesian quality criterion
data vector

A pdf version (484kB) is available.
The report partially builds on these publications: