Markov Combinations of Discrete Statistical Models

Markov combination is an operation that takes two statistical models and produces a third whose marginal distributions include those of the original models. Building upon and extending existing work in the Gaussian case, we develop Markov combinations for categorical variables and their statistical models. We present several variants of this operation, both algorithmically and from a sampling perspective, and discuss relevant examples and theoretical properties. We describe Markov combinations for special models such as regular exponential families, discrete copulas, and staged trees. Finally, we offer results about model invariance and the maximum likelihood estimation of Markov combinations.