Product Space and Coupling

Abstract

Coupling is a method for combining two random variables defined on separate probability spaces in order to study them simultaneously. It is the reverse process of marginalization, which separates a joint probability distribution into its marginal distributions. Coupling is particularly useful when we have two independent stochastic processes that we wish to analyze together. The Monge and Kantorovich problems are optimization problems that involve finding couplings between probability distributions, and they have been used in a variety of fields including economics, statistics, and machine learning. The product measure is a baseline coupling that assumes the two random variables are independent by construction. Under the product measure, the Tonelli–Fubini theorem can be formulated, which allows for the interchange of integration with respect to multiple variables.