Bayesian networks

informatyka
matematyka

monograficzne

This course studies graphical models, a subject that is an interaction between probability theory and graph theory. The topic provides a natural tool for dealing with a large class of problems containing uncertainty and complexity. These features occur throughout applied mathematics and engineering and therefore the material treated has diverse applications in the engineering sciences.

Course Content

Conditional independence, graphs and d- separation, Bayesian

networks

Markov equivalence for graph structures, the essential graph.

Causality and Intervention Calculus

Evidence: hard evidence, soft evidence, virtual evidence,

Jeffrey’s rule and Pearl’s method of Virtual Evidence.

Parametrising the network and sensitivity to parameter changes.

Model building and using computer software.

Decomposable graphs, junction trees and probility updating.

Factor graphs and the sum product algorithm

Bayesian inference, multinomial sampling and the Dirichlet integral.

Learning the conditional probability potentials for a given graph

structure.

Learning the graph structure.

By the end of the course, the student should:

1) understand the concept of factorising a distribution along a directed acyclic graph and what d-separation properties in the graph indicate about conditional independence.

2) understand the use of a directed acyclic graph to describe a causal network and the basics of Pearl's intervention calculus, how

interventions may be expressed in terms of sugery on the graph and how to compute conditional probabilities when the conditioning is by intervention.

3) understand how to produce a junction tree from a directed acyclic graph and use it to update a probability distribution when evidence is inserted (the Aalborg algorithm).

4) understand the principles of the Bayesian update for the conditional probabilities for a network with given graph structure, given complete instantiations, incomplete instantiations and fading.

5) understand the principles of constraint based and of search and score structure learning techniques, with particular emphasis on learning the essential graph. For constraint based techniques, the Recursive Autonomy Identification algorithm will be considered in detail, while for search and score, the principles of Monte carlo search techniques will be considered.

Assessment will be based on tutorial work and seminars based on the recent literature.