Bayesian Networks  An
Introduction
Henrik Bengtsson
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
Lund University
1999:E15

Abstract:

The report covers the basic concepts and theory of Bayesian Networks, which
are graphical models for reasoning under uncertainty. The graphical presentation
makes them very intuitive and easy to understand, and almost any person,
with only limited knowledge of Statistics, can for instance use them for
decision analysis and planning. This is one of many reasons to why they are
so interesting to study and use.
A Bayesian network can be thought of as a compact and convenient way to represent
a joint probability function over a finite set of variables. It contains
a qualitative part, which is a directed acyclic graph where the vertices
represent the variables and the edges the probabilistic relationships between
the variables, and a quantitative part, which is a set of conditional probability
functions.
Before receiving new information (evidence), the Bayesian network represents
our {a priori} belief about the system that it models. Observing the state
of one of more variables, the Bayesian network can then be updated to represent
our a posteriori belief about the system. This report shows a technique how
to update the variables in a Bayesian network. The technique first compiles
the model into a secondary structure called a junction tree representing
joint

distributions over nondisjoint sets of variables. The new evidence is inserted,
and then a {message passing} technique updates the joint distributions and
makes them consistent. Finally, using marginalization, the distributions
for each variable can be calculated. The underlying theory for this method
is also given.


All necessary algorithms for implementing a basic Bayesian network application
are presented along with comments on how to represent Bayesian networks on
a computer system. For validation of these algorithms a Bayesian network
application in Java was implemented.


Keywords:

Bayesian networks, belief networks, junction tree algorithm, probabilistic
inference, probability propagation, reasoning under uncertainty.