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

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
 Keywords
 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:
 D. Heckerman, D. Geiger, D.M. Chickering:
Learning Bayesian networks: the combination of knowledge and statistical data. Machine Learning 20 (1995) pp. 194243.
 M. Studeny:
Probabilistic Conditional Independence Structures. SpringerVerlag, London, 2005.