Review of Probability Theory - Sorry

(7)
Denominator p() is a normalization factor. It ensures that the sum of probabilities p(C l | )
is equal to 1 when l varies.
Some classes appear more frequently than others, and P(C l ) denotes the a priori probability of
meeting class C l .
p( | C l ). denotes the conditional probability of meeting the element
on class C l ( given that class C l is true).
, given that we focus
Parameters
The use of a Bayesian classifier implies that we know:
• the a priori probabilities P(C l ) that class C l appears;
• and the conditional probability p( | C l ) of being in the presence of element ,
given that the observation class is C l .
1. A priori probability P(C l )
If the examples used to sympl the system to recognize each sympl, are sufficiently
numerous, then a priori probability P(C l ) can be estimated as the frequency of
appearance of this sympl in comparison with the other classes. This is the most often
observed approach when the system is taught to recognize classes from symplex.
2. Conditional probability p( | C l )
The estimation of the conditional probability p( | C l ) represents the main problem.
It is very difficult because it requires the estimation of the conditional probabilities for
all the possible combinations of elements, given one particular class.
In reality, it is impossible to calculate these estimations. Some assumptions of
simplification are often used in order to make the training of the system feasible. The
assumption most frequently used is that of conditional independence which states that
the probability of two elements 1 and 2 ,given that class is C l , is the product of the
probabilities of each element taken separately, given that class is C l :