Vapnik's book talks about Fisher's discriminant analysis (which we commonly call pattern recognition) where Fisher proposes the model: There exist two categories of data distributed according to two different statistical laws $p_1(x,\alpha^*)$ and $p_1(x,\beta^*)$ (densities, belonging to parametric classes). Let the probability of occurrence of the first category of data be $q_i$ and the probability of the second category be $(1 - q_i)$. The problem is to find a decision rule that minimizes the probability of error. It says that The smallest probability of error is achieved by the decision rule that considers vector x as belonging to the first category if the probability that this vector belongs to the first category is not less than the probability that this vector belongs to the second category. This happens if the following inequality holds:
$q_i*p_1(x,\alpha^*) \geq (1 - q_i)*p_1(x,\beta^*).$
What is the reason for this?