Optimal Poisson Cognitive System with Markov Learning Model

The aim of the study is to develop a mathematical model of the trained Markov cognitive system in the presence of discrete training and interfering random stimuli arising at random times at its input. The research method consists in the application of the simplest Markov learning model of Estes with a stochastic matrix with two states, in which the transition probabilities are calculated in accordance with the optimal Neуman-Pearson algorithm for detecting stimuli affecting the system. The paper proposes a model of the random appearance of images at the input of the cognitive system (in terms of learning theory, these are stimuli to which the system reacts). The model assumes an exponential distribution of the system’s response time to stimuli that is widely used to describe intellectual work, while their number is distributed according to the Poisson law. It is assumed that the cognitive system makes a decision about the presence or absence of a stimulus at its input in accordance with the Neуman-Pearson optimality criterion, i.e. maximizes the probability of correct detection of the stimulus with a fixed probability of false detection. The probabilities calculated in this way are accepted as transition probabilities in the stochastic learning matrix of the system. Thus, the following assumptions are accepted in the work, apparently corresponding to the behavior of the system assuming human reactions, i.e. the cognitive system.
The images analyzed by the system arise at random moments of time, while the duration of time between neighboring appearances of images is distributed exponentially.
The system analyzes the resulting images and makes a decision about the presence or absence of an image at its input in accordance with the optimal Neуman-Pearson algorithm that maximizes the probability of correct identification of the image with a fixed probability of false identification.
The system is trainable in the sense that decisions about the presence or absence of an image are made sequentially on a set of identical situations, and the probability of making a decision depends on the previous decision of the system.
The new results of the study are analytical expressions for the probabilities of the system staying in each of the possible states, depending on the number of steps of the learning process and the intensities of useful and interfering stimuli at the input of the system. These probabilities are calculated for an interesting case in which the discreteness of the appearance of stimuli in time is clearly manifested and the corresponding graphs are given. Stationary probabilities are also calculated, i.e. for an infinite number of training steps, the probabilities of the system staying in each of the states and the corresponding graph is presented.
In conclusion, it is noted that the presented graphs of the behavior of the trained system correspond to an intuitive idea of the reaction of the cognitive system to the appearance of stimuli. Some possible directions of further research on the topic mentioned in the paper are indicated.

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