In the past few years, probabilistic couplings have been identified as a powerful abstraction for verifying probabilistic properties. Originating from research in probability theory, a coupling is a distribution over pairs that relates—or couples—two given distributions. If we can find a coupling with certain properties, then we can conclude properties about the two related distributions. In this way, probabilistic relational properties—properties comparing two executions of a probabilistic program—can be established by building a suitable coupling.

Couplings have also been explored in the logic and verification literature. For example, probabilistic bisimulation assert that there exists a coupling; in this way, couplings can be used to verify equivalence of finite state probabilistic transition systems. However, their use in mathematics suggests that couplings can prove more sophisticated properties for richer probabilistic computations, such as imperative programs and infinite state systems. Furthermore, we can borrow a tool from probability theory called proof by coupling to construct couplings in a compositional fashion.

This tutorial will survey recent techniques in program verification for building and reasoning about probabilistic couplings. These methods have enabled the first formal verification of probabilistic algorithms and properties that previously seemed out of reach. Target properties range from program equivalence and probabilistic independence to more complex guarantees, such as convergence of random walks and differential privacy. This tutorial will first give participants and intuitive feel for how to apply proofs by coupling, for formal or informal reasoning, and then demonstrate how this technique can be formalized in logic.