Conditional Bayesian task of testing many hypotheses

Conditional Bayesian task of testing many hypotheses is stated and solved. The concept of conditionality is used for the designation of the fact that the Bayesian task is stated as a conditional optimization problem where the probability of one-type error is restricted and, under such a condition, the probability of second-type error is minimized. The offered statement gives the decision rule which allows us not to accept any hypothesis if, on the basis of the available information, it is impossible to make a decision with the set significance level. In such a case, it is necessary to ensure the additional information in the form of additional observation results or a change in the significant level of hypotheses testing. These properties make our statement more general than the usual statement of the Bayesian problem which is a special case of the one offered and improve the reliability of the made decision. The calculation results completely confirm the results of theoretical investigations.