Bernal S D_2010.pdf - University of Plymouth

3.3. DEFINITION AND MATHEMATICAL FORMULATION
In this case it can be said that set 2 separates sets X and ?. written as X ±L Y\Z.
Factorization From the definition of condilional probability it follows that the joint probabil­
ity distribution of a set of hierarchically organized variables X can be faclorized as follows (also
referred to as the chain rule),
P{Xu...,X„)=PiXi\x2 X„)-P{X2\Xi....X„)-...-P{x„)=llP(Xi\x.+ ,....,X„) (3.7)
The different condilional probability terms can be simplified according to conditional indepen­
dence assumptions.
Baycs theorem Given two sets X and ?, the conditional probability of Y given )( (also called
the posterior probability) satisfies the following equation,
where the conditional probability P{x\y) is also called the liki-lihuoii; the marginal probability
P{y) is also called \hepnor; and the marginal probability P{x) acts as a narmalizaliun constant.
The marginalization and factorization of the joint probability disiribiilion, together with the
application of the Bayes theorem, are the three key elements of the belief propagation algorithm
described in the following section.
3.3.2 Bayesian networks
A Bayesian network is a specific type of graphical model, more specilically a directed acyclic
graph, where each node in the network represents a random variable, and arrows establish a
causal dependency between nodes. Therefore, each arrow represents a condilional probability
distribution P{X\nx) which relates node X with its parents Uy- Crucially, the network is de­
fined such thai the probability of a node X being in a particular slate depends only on the slate
of its parents. Fix- Consequently, a Bayesian network of N random variables X, defines a joint
probability distribution which can be faclorized as follows,
76
I