In the Discrete Sense
For pattern recognition tasks, many times the data
that is required to be analyzed and classified is a discrete vector of
(sometimes assumed independent) components: X = (x1,x2,x3,...,xn).
This vector is usually found by some type of pattern recognition
algorithms. To make a decision on a feature vector in the discrete
case, we must first investigate Bayes rule in discrete form.
As was seen in the continuous case of Bayes rule,
the discrete case is much similar. However, instead of the feature
vector x being
a point in d-dimensional space, it now assumes one of m
discrete values: v1,...,vm.
Consequently, integrals are therefore replaced with summations:
Furthermore, instead of utilizing probability
density functions as was used if the random variable was
continuous, the distribution is now a probability distribution or just
a probability. Hence:
Nonetheless, as will be seen, the Bayes decision
rule remains unchanged as it’s purpose is: to minimize the
risk or cost in the decision.
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