Bayesian adaptation in Poisson cognitive systems

The aim of the study is to investigate the possibility of applying Bayesian adaptation algorithms to cognitive systems that perceive the Poisson process of external events.

The method of research is the use of stochastic description and synthesis of cognitive systems, including the theory of doubly stochastic Poisson processes and the theory of Bayesian adaptation. The formal definition of cognitive systems in the state space in the spirit of similar definitions of the theory of dynamic systems is formulated. The definition has become a methodological basis for the development of models of those sets and transformations that are characteristic of cognitive systems. In particular, to describe the stochastic properties of cognitive systems and the possibility of creating an optimal algorithm, the Bayesian approach recognized in a number of philosophical works is applied.

The optimal estimate by the criterion of the minimum standard error is, as is known, a posteriori mathematical expectation of a random estimated value, which is applied in this work. In this case, the well- known difficulty of using Bayesian optimal estimation is the need to set a priori probabilities of a random variable in the system under consideration. An adaptive Bayesian estimation algorithm, also known as the empirical Bayesian approach, is used to overcome this problem. According to the above it is believed that at the entrance of the cognitive system, namely in the unconscious in continuous time there are some events that are modeled by random points. The intensity of the appearance of points is determined by a random variable X, the evaluation of which is the task of the cognitive system as a whole. Up to some time in the field of the unconscious the number of random events accumulate (in mathematical language the classifying sample is formed). At some point, an attempt is made to estimate the value of X, i.e. an attempt to move information from the unconscious area of the cognitive system to the conscious, which is a mental act, an act of learning, etc. From a mathematical point of view, such a model of cognitive functioning is the implementation of an adaptive Bayesian approach, which allows to reduce the influence of a priori distribution of an unknown quantity on its evaluation.

The described model of the cognitive system is justified by the fact that the value of X is not only random, but also with an unknown a priori distribution, is not observed directly, and in some way must be evaluated by the cognitive system on the basis of the already existing in the unconscious number of events and the last event on the basis of which.

The optimal estimation of the random parameter is used to solve the problem of classification of observations, i.e. the optimal verification of the one-sided hypothesis by the Bayesian criterion.

As a result of the undertaken consideration the applicability of the developed formal definition of cognitive system for the formulation of various problems of analysis and synthesis of systems is demonstrated. The advantage of the applied model is the minimum amount of a priori information about the processes occurring in the system. One assumption about the Poisson nature of the events occurring at the input of the system was sufficient.

The results of a computational experiment on the adaptive estimation of a random parameter with an unknown a priori distribution are presented.

In conclusion it is noted that the further development of the study can be a detailed formulation of the mathematical properties of the elements of the cognitive system mentioned in the definition, formulation, solution and interpretation of new mathematical problems of analysis and synthesis.

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