Titel_SS06

BC D

A

BC D

A

Figure 10.4: Illustration of a converging network.

Formally a Bayesian network is composed of:

 A set of variables and a set of directed edges (or connections) between the variables.

 Each variable may have a countable or uncountable set of mutually exclusive states.

 The variables together with the directed edges form a directed a-cyclic graph (DAG).

 To each variable A with parents B, C, D, ..etc. there is assigned a conditional probability
structure PABCD( , , ,..).

In case the variable A has no parents the conditional probability structure reduces to the
unconditional probability of A, i.e. PA().

10.4 BPN’s with Discrete State Variables

It is possible to work with Bayesian nets containing variables with continuous states as well as
discrete states but in general for mathematical reasons it often becomes necessary to discretize
continuous state variables into discrete state variables. For this reason the following considers
Bayesian networks where the variables can only attain discrete states.

Assume that all n variables AA A 12 ,,..nof a Bayesian network are collected in the vector

(, ,.. 12 n)

U A AAT, also called the universe. In general it is of interest to be able to assess the

joint probability distribution of the universe i.e. PPAAA() ( , ,.. )U  12 n , any marginalized set

of the universe as well as to assess such probability distributions subject to evidence

in regard to the states of individual variables, e.g.

PA( i)

e PAe()i.

A Bayesian Network can be considered to be a special representation of such probability
distributions and using the so-called chain rule of probability calculus it is possible to write
the probability distribution functionP()U in the following form:

() (ii( ))

i

PPApaU . A (10.1)

where ( )paAi is the parent set of the variable. The probability distribution function for

elements of U, e.g. for can be achieved by marginalization i.e.

Ai

Aj

\\

() () ( ())

jj

j

AAi

PAP .PA paA

UU

U ii (10.2)