  In this work we take a Category Theoretic perspective on the relationship
between probabilistic modeling and function approximation. We begin by defining
two extensions of function composition to stochastic process subordination: one
based on the co-Kleisli category under the comonad (Omega x -) and one based on
the parameterization of a category with a Lawvere theory. We show how these
extensions relate to the category Stoch and other Markov Categories. Next, we
apply the Para construction to extend stochastic processes to parameterized
statistical models and we define a way to compose the likelihood functions of
these models. We conclude with a demonstration of how the Maximum Likelihood
Estimation procedure defines an identity-on-objects functor from the category
of statistical models to the category of Learners. Code to accompany this paper
can be found at
https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood
