  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 within quantum field theory. Here,
we will discuss how discretized Euclidean field theories, such as the
$\phi^{4}$ lattice field theory on a square lattice, are mathematically
equivalent to Markov fields, a notable class of probabilistic graphical models
with applications in a variety of research areas, including machine learning.
The results are established based on the Hammersley-Clifford theorem. We will
then derive neural networks from quantum field theories and discuss
applications pertinent to the minimization of the Kullback-Leibler divergence
for the probability distribution of the $\phi^{4}$ machine learning algorithms
and other probability distributions.
