Recognition with unknown prior probabilities of images

may be unknown for many reasons, in particular, if they are unknown functions of time or some uncontrollable circumstances, conditions. In this case, the Bayes decision rule cannot be used. Instead of the risk of loss (in the particular case of the average probability of recognition errors), we have to deal with a vector (in the particular case ).

The task is set as follows: from all possible sets under conditions and it is necessary to choose such (and further use it with recognition), when the maximum component of the vector is minimal.

Algorithmically, one of the simplest is the Monte Carlo method. Randomly times vectors are selected The vector at which the maximum component takes the smallest value accepted for use. The more , the higher the probability of "hitting" the nearest neighborhood of the optimal vector . Of course, complete enumeration of options is possible, but it is acceptable only if there is not a very large number of possible . In some particular tasks, an analytical approach to searching can be implemented. . Consider the case with two images. (Fig. 21).

Fig. 21. Scope of the problem of definition
by minimax criterion

The solution of the minimax problem lies on the line segment Denote by the object area of ​​the first image, and through - the second. It's clear that The average probability of recognition errors is determined by Build a graph (Fig. 22).

It's obvious that at and = 1. Between them are values at which including its maximum value. Let's say we chose = . Then as a function of true (but unknown) value lies on a straight line tangent to at the point corresponding to = . Moreover, if the true value lies to the left of the point (eg, = ), then the actual average recognition error ( ) will be less than predicted at = . But if the true value = , the actual average error ( ) will be significantly more predictable. Similar reasoning can be given for the right slope of the curve by putting for example = . Only by choosing = that corresponds to the maximum curve we guarantee that will not exceed whatever the true meaning .

Fig. 22. Dependence of the probability of recognition error
from

Consider the analytical formulation of the problem of finding a minimax solution (it should be borne in mind that and depends on since they are functions from and , and the latter depend on the prior probabilities of the images).

Denote through . Need to find such a value in which

Where .

From this equation it is clear that to find its analytical solution is very difficult. First, you need to write explicitly the dependency. and from and secondly, the equation must have an analytical solution. In the simplest cases, this is possible, but the simplest cases, unfortunately, are extremely rare in practice.