The paper by J. Močkus discusses Bayesian methods for seeking the extremum in global optimization problems. The author introduces the concept of Bayesian optimization, which is based on minimizing the expected deviation from the extremum. The method is defined in the context of a stochastic function \( f(x) = \int(x, \omega) \), where \( \omega \) is an unknown index. The probability distribution \( P \) on \( \Omega \) is determined by the a priori probability distribution function \( F_{x_i, \ldots, x_n}(y_1, \ldots, y_n) \). The decision function \( d^* \) is derived to minimize the expected deviation from the extremum, and it is shown that this can be achieved through a set of recurrent equations. The paper also addresses the case of noisy observations and provides an illustrative example to demonstrate the method's application. Convergence conditions for the Bayesian method are established, including assumptions about the compactness of the set \( A \), continuity of functions, and specific properties of the conditional probability distribution functions. The one-stage method, a simplified version of the Bayesian method, is introduced, where each observation is assumed to be the last one. This method converges under the same conditions as the full Bayesian method. The paper also discusses the restricted-memory case, where only a limited number of observations can be remembered, and notes that this method may not always converge. Finally, the implementation of the one-stage Bayesian method for specific types of functions, such as Gaussian fields and Wiener processes, is described, and the development of suitable a priori distributions is highlighted as a key challenge in practical applications.The paper by J. Močkus discusses Bayesian methods for seeking the extremum in global optimization problems. The author introduces the concept of Bayesian optimization, which is based on minimizing the expected deviation from the extremum. The method is defined in the context of a stochastic function \( f(x) = \int(x, \omega) \), where \( \omega \) is an unknown index. The probability distribution \( P \) on \( \Omega \) is determined by the a priori probability distribution function \( F_{x_i, \ldots, x_n}(y_1, \ldots, y_n) \). The decision function \( d^* \) is derived to minimize the expected deviation from the extremum, and it is shown that this can be achieved through a set of recurrent equations. The paper also addresses the case of noisy observations and provides an illustrative example to demonstrate the method's application. Convergence conditions for the Bayesian method are established, including assumptions about the compactness of the set \( A \), continuity of functions, and specific properties of the conditional probability distribution functions. The one-stage method, a simplified version of the Bayesian method, is introduced, where each observation is assumed to be the last one. This method converges under the same conditions as the full Bayesian method. The paper also discusses the restricted-memory case, where only a limited number of observations can be remembered, and notes that this method may not always converge. Finally, the implementation of the one-stage Bayesian method for specific types of functions, such as Gaussian fields and Wiener processes, is described, and the development of suitable a priori distributions is highlighted as a key challenge in practical applications.