Foundations of Data Science

By iterating the above formulas we can arrive at a local optimum for the HMM parameters
a i,j and b j (O k ).
9.3 Graphical Models, and Belief Propagation
A graphical model is a compact representation of a function of n variables x 1 , x 2 , . . . , x n .
It consists of a graph, directed or undirected, whose vertices correspond to variables that
take on values from some set. In this chapter, we consider the case where the function
is a probability distribution and the set of values the variables take on is finite, although
graphical models are often used to represent probability distributions with continuous variables.
The edges of the graph represent relationships or constraints between the variables.
The directed model represents a joint probability distribution that factors into a product
of conditional probabilities.
p (x 1 , x 2 , . . . , x n ) =
n∏
p (x i |parents of x i )
i=1
It is assumed that the directed graph is acyclic. The directed graphical model is called
a Bayesian or belief network and appears frequently in the artificial intelligence and the
statistics literature.
The undirected graph model, called a Markov random field, can also represent a joint
probability distribution of the random variables at its vertices. In many applications the
Markov random field represents a function of the variables at the vertices which is to be
optimized by choosing values for the variables.
A third model called the factor model is akin to the Markov random field, but here the
dependency sets have a different structure. In the following sections we describe all these
models in more detail.
9.4 Bayesian or Belief Networks
A Bayesian network is a directed acyclic graph where vertices correspond to variables
and a directed edge from y to x represents a conditional probability p(x|y). If a vertex x
has edges into it from y 1 , y 2 , . . . , y k , then the conditional probability is p (x | y 1 , y 2 , . . . , y k ).
The variable at a vertex with no in edges has an unconditional probability distribution.
If the value of a variable at some vertex is known, then the variable is called evidence.
An important property of a Bayesian network is that the joint probability is given by the
product over all nodes of the conditional probability of the node conditioned on all its
immediate predecessors.
In the example of Fig. 9.1, a patient is ill and sees a doctor. The doctor ascertains
the symptoms of the patient and the possible causes such as whether the patient was in
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