Coupling is a term used in various contexts, including software engineering, physics, biology, and engineering. Here are a few of its meanings in different fields: 1. **Software Engineering**: In the context of software architecture and design, coupling refers to the degree of dependence between software modules or components. Low coupling is desirable because it means that changes in one module are less likely to require changes in another, leading to more maintainable and flexible codebases.
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A coupling of probability distributions is a joint probability distribution of random variables whose marginal distributions are . It lets us choose the statistical dependence between two random constructions while preserving the individual laws. This choice can make a comparison much easier: giving two simulations the same random input may keep their paths close, even though separately generating both paths would hide that relationship.[ref]
Markov Chains and Mixing Times, second edition by David A. Levin and others. §4.2, coupling definition and Example 4.6, printed pp. 49–50
For example, two fair coin tosses can be generated independently, with mismatch probability , or by tossing once and setting , with mismatch probability zero. Both constructions have the same two Bernoulli distributions as marginals. Their joint laws differ. A maximal coupling makes the values agree as often as possible; a coalescing coupling arranges permanent agreement of two processes after a random time. Minimizing expected geometric separation instead leads to Wasserstein distance.