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The book is dedicated to the solution of the Bayesian problem of many hypotheses testing with respect to the parameters of multivariate normal distribution and derivation of working formulas for simple testing of many hypotheses in a limited period of time, and calculation of the suitable value of the risk function, which is the objective criterion of the quality of the decision made. Also a new sequential method of testing many hypotheses based on special properties of regions of acceptance of hypotheses in the conditional Bayesian task of testing many hypotheses is formalized.The results of sensitivity analysis of the considered Bayesian problem are given; the results of experimental research of the developed optimal, quasi-optimal and sequential Bayesian methods are presented.