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Recognition with unknown prior probabilities of images

Lecture



  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   Recognition with unknown prior probabilities of images ).

Minimax criterion

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

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

  Recognition with unknown prior probabilities of images

Fig. 21. Scope of the problem of definition   Recognition with unknown prior probabilities of images
by minimax criterion

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

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

  Recognition with unknown prior probabilities of images

Fig. 22. Dependence of the probability of recognition error
from   Recognition with unknown prior probabilities of images

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

Denote   Recognition with unknown prior probabilities of images through   Recognition with unknown prior probabilities of images . Need to find such a value   Recognition with unknown prior probabilities of images in which

  Recognition with unknown prior probabilities of images

Where   Recognition with unknown prior probabilities of images   Recognition with unknown prior probabilities of images .

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


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Pattern recognition

Terms: Pattern recognition