Remember that classification is a supervised learning concept that has to do with determining the the class a new input variable belongs.

In trying to assign input variables to classes, there is possibility that we may assign to a wrong class. This is called misclassification.

By definition, misclassifcation occurs when an input variable is assigned to the wrong class.

One of the goals of classification is to minimize the number of misclassifications. This is done by defining a rule that assigns input x to one of the available classes.

The approach is to divide the input space into regions

Consider the case of two classes C

We can represent this as follows:

To minimize misclassification, we must choose to assign x to which of the classes has the smaller value of the integrand.

So if

Using the product rule, we can determine the posterior probability:

Note that the term

For the more general case of K classes, it would be a bit easier to maximize the probability of being correct and this is given by:

This means the to minimize misclassification, we need to maximize this probability over the region

Using the product rule which states that:

We can see that each class has to be assigned to the class that have the highest posterior probability

In trying to assign input variables to classes, there is possibility that we may assign to a wrong class. This is called misclassification.

By definition, misclassifcation occurs when an input variable is assigned to the wrong class.

One of the goals of classification is to minimize the number of misclassifications. This is done by defining a rule that assigns input x to one of the available classes.

The approach is to divide the input space into regions

*R*called decision regions, one region for each class._{k}*R*is assigned to class_{k}*C*_{k}Consider the case of two classes C

_{1}and C_{2}. A mistake occurs when an input vector belonging to R_{1}is assigned to C_{2}or vector x belonging to R_{2}is assigned to C_{1}.We can represent this as follows:

So if

*p(x, C*is greater than_{1})*p(x, C*, then x would be assigned to C_{2})_{1}.Using the product rule, we can determine the posterior probability:

*p(x, C*

_{k}) = p(C_{k}| x)p(x)Note that the term

*p(C*is known as the posterior probability and x should be assigned to the class having the largest posterior probability_{k}| x)*p(C*_{k}| x).For the more general case of K classes, it would be a bit easier to maximize the probability of being correct and this is given by:

*R*._{k}Using the product rule which states that:

*p(*

**x**

*, C*

_{k}) = p(C_{k}|**x**

*)p(x)*

We can see that each class has to be assigned to the class that have the highest posterior probability

*p(C*_{k}|**x***)*