A pre-expectation calculus for probabilistic sensitivity

Sensitivity properties describe how changes to the input of a program affect output, typically by upper bounding distance between outputs two runs monotone function corresponding inputs. When programs are probabilistic, is distributions. The Kantorovich lifting provides general way defining distributions underlying sample space; choosing an appropriate on base space, one can recover other usual probabilistic distances, such as Total Variation distance. We develop relational pre-expectation calculus bound executions program. illustrate our methods proving algorithmic stability machine learning algorithm, convergence reinforcement and fast mixing for card shuffling algorithms. also consider some extensions: using show Markov chains uniform distribution over states asynchronous extension reason about pairs with different control flow.