Bayesian network are graphical models for joint probability distribution.
Consider to model the inheritance of a single gene that determines a person's blood type:
Blood Type Domain
| The blood type domain is a genetic model of the inheritance of a single gene that determines a person's blood type. Each person has two copies of the chromosome containing this gene, one inherited from her mother and one inherited from her father. Occasionally, a person is unavailable for testing, and yet because of the clarification of crime, test of paternity, allocation of (frozen) semen etc. it is often necessary that a blood type of the person be estimated. A blood type can still be derived for that person through an examination and analysis of the types of family members. |
More formally, a Bayesian network is an augmented, directed acyclic graph, where
each node corresponds to a random variable xi and each edge indicates a direct
influence among the random variables. It represents the joint probability
distribution
P(x1,..., xn) |
over a fixed, finite set {x1,,...,xn} of random variables.
Each random variable xi possesses a finite set S(xi) of mutually exclusive
states. The following Figure shows the graph of a Bayesian network modelling the blood
type example for a particular family:
The familial relationship, which is taken
from Jensen's stud farm example, forms the basis for the graph. The network
encodes e.g. that Dorothy's blood type is influenced by the genetic information
of her parents Ann and Brian. The set of possible states of bt(dorothy)
is
S(bt(dorothy)) = {a, b, ab, 0};
|
the set of possible states of pc(dorothy) and
mc(dorothy) are
S(pc(dorothy)) = S(mc(dorothy)) = {a, b, 0}.
|
The same holds
for ann and brian. The direct predecessors of a node x, the parents of x are denoted
by Pa(x). For instance,
Pa(bt(ann)) = {pc(ann), mc(ann)}.
|
A Bayesian network stipulates the following conditional independence assumption
Independece Assumption
Each node xi in the graph is conditionally independent of any subset A of nodes
that are not descendants of xi given a joint state of Pa(xi), i.e.
P(xi | A, Pa(xi)) = P(xi | Pa(xi)). |
For example, bt(dorothy) is conditionally independent of bt(ann) given a joint
state of its parents {pc(dorothy), mc(dorothy)}. Any pair (xi, Pa(xi)) is called the
family of xi denoted as Fa(xi), e.g. bt(dorothy)'s family is
(bt(dorothy), {pc(dorothy), mc(dorothy)}).
Because of the conditional independence assumption, the joint
probability density factorizes
P(x1,...,xn) = Prod P(xi|Pa(xi)) |
Thereby, we associate with each node xi of the graph the conditional probability distribution P(xi | Pa(xi)), denoted as cpd(xi).