The misclassification rate is the probability of a data point being classified into the wrong class.
Minimizing the misclassification rate involves assigning each point to the class for which the posterior probability is highest.
For the two-class case, the misclassification rate is given by:
where R1 is the region of input space assigned to class C1 and R2 is the region assigned to class C2.
To minimize this, we should assign each value of x to the class for which p(x,Ck) is smaller. Since p(x,Ck)=p(Ck∣x)p(x), and the factor p(x) is common to both terms, we can minimize the misclassification rate by assigning each value of x to the class for which the posterior probability p(Ck∣x) is largest.