Quantum field theories, Markov random fields and machine learning

The transition to Euclidean space and the discretization of quantum field theories on spatial or space-time lattices opens up the opportunity to investigate probabilistic machine learning from the perspective of quantum field theory. Here, we will discuss how discretized Euclidean field theories can be recast within the mathematical framework of Markov random fields, which is a notable class of probabilistic graphical models with applications in a variety of research areas, including machine learning. Specifically, we will demonstrate that the $\phi^{4}$ scalar field theory on a square lattice satisfies the Hammersley-Clifford theorem, therefore recasting it as a Markov random field from which neural networks are additionally derived. We will then discuss applications pertinent to the minimization of an asymmetric distance between the probability distribution of the $\phi^{4}$ machine learning algorithms and that of target probability distributions.