An Advanced Probabilistic Framework for Assisting Screening Mammogram Interpretation - Computational Intelligence in Healthcare: Advanced Methodologies - page 366

In many practical applications, parameter estimation for naive Bayes models
uses the method of maximum likelihood estimates. Despite their naive design and
apparently over-simplified assumptions, naive Bayes classifiers often demonstrate
a good performance in many complex real-world situations.
The naive Bayes classifier combines the above naive Bayesian network model
with a decision rule [27]. A common rule is choosing most probable hypothesis-a
maximisation criterion. In this case, the classifier is defined by the function:

P ( Cl = c ) n NB( f 1 ,...,f n ) = argmax

c

P ( F i = f i | Cl = c ) . (1)

i =1

4.3 Causal Independence

It is known that the number of probabilities in a CPT for a certain variable
grows exponentially in the number of parents in the ADG. Therefore, it is often
infeasible to define the complete CPT for variables with many parents. One
way to specify interactions among statistical variables in a compact fashion is
offered by the notion of causal independence [28], where multiple causes (parent
nodes) lead to a common effect (child node). The general structure of a causal-
independence model is shown in Figure 3 and below we give a brief description
of the notion following the definitions in [29].

C 1 C 2 ... C n

I 1 I 2 ... I n

f E

Fig. 3. Causal-independence model

A causal-independence model expresses the idea that causes C 1 ,...,C n influ-
ence a given common effect E through intermediate variables I 1 ,...,I n .Wede-
note the assignment of a value to a variable by lower-case letter, e.g., i k stands for
I k = ( true )and ı k otherwise. The interaction function f represents in which

way the intermediate effects I k , and indirectly also the causes C k ,interact.This
function f is defined in such way that when a relationship between the I k 's
and E =

is satisfied, then it holds that f ( I 1 ,...,I n )= e ;otherwise,itholds
that f ( I 1 ,...,I n )= e . Furthermore, it is assumed that if f ( I 1 ,...,I n )= e then
P ( e I 1 ,...,I n )=0.

Using information from the topology of the network, the notion of causal inde-
pendence can be formalised for the occurrence of effect E , i.e. E =

| I 1 ,...,I n ) = 1; otherwise, if f ( I 1 ,...,I n )= e ,then P ( e |

,interms

of probability theory by: