Page 138 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Introduction to Artificial Intelligence & Expert Systems

Notes 3. At a high level, our goal is to efficiently represent and operate upon a joint distribution P
r over a set of random variables X = fX1;:::; Xng. Even if these variables are binary-valued,
a naive representation of the joint distribution requires the specification of 2 n numbers
(the probabilities of the 2 n different assignments to the variables), which would be
infeasible except for very small n. Fortunately, most real-world application domains
exhibit a high degree of structure in this joint distribution that allows us to factor the
representation of the distribution into modular components.

7.1.4 Graphical Models for Uncertain Reasoning

A Bayesian network, Bayes network, belief network, Bayes(ian) model or probabilistic directed
acyclic graphical model is a probabilistic graphical model (a type of statistical model) that
represents a set of random variables and their conditional dependencies via a directed acyclic
graph (DAG). For example, a Bayesian network could represent the probabilistic relationships
between diseases and symptoms. Given symptoms, the network can be used to compute the
probabilities of the presence of various diseases.
Formally, Bayesian networks are directed acyclic graphs whose nodes represent random variables
in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters
or hypotheses. Edges represent conditional dependencies; nodes which are not connected
represent variables which are conditionally independent of each other. Each node is associated
with a probability function that takes as input a particular set of values for the node’s parent
variables and gives the probability of the variable represented by the node. For example, if the
parents are m Boolean variables then the probability function could be represented by a table of
m
m
2 entries, one entry for each of the 2 possible combinations of its parents being true or false.
Similar ideas may be applied to undirected, and possibly cyclic, graphs; such are called Markov
networks.
Efficient algorithms exist that perform inference and learning in Bayesian networks. Bayesian
networks that model sequences of variables (e.g. speech signals or protein sequences) are called
dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve
decision problems under uncertainty are called influence diagrams.

Inference, or model evaluation, is the process of updating probabilities of outcomes based upon
the relationships in the model and the evidence known about the situation at hand. When a
Bayesian model is actually used, the end user applies evidence about recent events or observations.
This information is applied to the model by “instantiating” or “clamping” a variable to a state
that is consistent with the observation. Then the mathematical mechanics are performed to
update the probabilities of all the other variables that are connected to the variable representing
the new evidence.

Notes After inference, the updated probabilities reflect the new levels of belief in (or
probabilities of) all possible outcomes coded in the model. These beliefs are mediated by
the original assessment of belief performed by the author of the model.

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