Probabilistic investigation of non-equilibrium neural network and decision-making models

The Prisoner's Dilemma game (PDG) is commonly used as an experimental tool to study the probabilistic aspects of human decision-making. Numerous behavioral tests have been undertaken over an extended period of time to investigate this game, revealing a consistent contravention of the well-established "sure thing principle." This concept holds significant importance within the rational theory of decision-making. The violation in question can be elucidated by quantum probabilistic models, which attribute it to a second-order interference phenomenon. This particular effect is outside the scope of explanation provided by classical probability theory. In this study, we utilize the framework of generalized probabilistic theories and analyze the phenomenon of interference from the perspective of quantum information theory in order to ascertain its origin. Specifically, we propose a modification to an established quantum probabilistic model by employing the density matrix formalism. We investigate varying levels of classical and quantum uncertainty pertaining to a player's anticipation of another player's action in the Prisoner's Dilemma Game (PDG). This allows us to illustrate that the presence of quantum coherence in the player's initial prediction and its conversion to probabilities during the dynamics is what enables the explanation of the violation. Furthermore, we explore the significance of additional quantum information-theoretical measures, such as quantum entanglement, in the process of decision-making. In conclusion, we put forth a novel expansion of the Probability Distribution Generator (PDG) framework, encompassing three distinct choices. This extension aims to facilitate a comprehensive evaluation of the predictive capabilities of quantum probability theory in comparison to a broader probabilistic theory that encompasses it as a specific instance while also demonstrating third-order interference phenomena. Physicists have been trying to use Fokker-Planck equations (FPE) in modeling the SGD training process. Through a mathematical model using FPE solutions for a stationary state in a noisy environment, we described the process of minibatching. One of our challenges during this project was constructing a simple, physical model which relates the minibatch noise to a general form of coordinate translation in the n-dimensional phase space to reproduce the observed stationary state. In my project, I showed that one could reproduce it by applying a simple shift in the space as source noise generated from a multivariate normal distribution. Our results demonstrate that the character of the noise determines whether the system ends up in a Boltzmannian or non-Boltzmannian stationary state. In addition, I used MNIST data set and trained them in Julia by minibatching to see how ML on an actual data set can result in stationary states predicted by our model. Indeed, reproducing the observed stationary state requires checking the properties of the covariance matrix, probability current, and diffusion matrix.