Tail Bound Analysis for Probabilistic Programs via Central Moments

For probabilistic programs, it is usually not possible to automatically derive exact information about their properties, such as the distribution of states at a given program point. Instead, one can attempt to derive approximations, such as upper bounds on tail probabilities. Such bounds can be obtained via concentration inequalities, which rely on the moments of a distribution, such as the expectation (the first raw moment) or the variance (the second central moment). Tail bounds obtained using central moments are often tighter than the ones obtained using raw moments, but automatically analyzing higher moments is more challenging. This paper presents an analysis for probabilistic programs that automatically derives symbolic over- and under-approximations for variances, as well as higher central moments. To overcome the challenges of higher-moment analysis, it generalizes analyses for expectations with an algebraic abstraction that simultaneously analyzes different moments, utilizing relations between them. The analysis is proved sound with respect to a trace-based, small-step model that maps programs to Markov chains. A key innovation is the notion of semantic optional stopping, and a generalization of the classical optional-stopping theorem. The analysis has been implemented using a template-based technique that reduces the inference of polynomial approximations to linear programming. Experiments with our prototype central-moment analyzer show that, despite the analyzer's over-/under-approximations of various quantities, it obtains tighter tail bounds than a prior system that uses only raw moments, such as expectations.