This Is the Moment for Probabilistic Loops

TL;DR

This paper introduces an automatic static analysis method for deriving higher moments of variables in probabilistic loops with uncountable state spaces, aiding in distribution recovery and tail probability computation.

Contribution

It presents a novel algebraic approach using linear recurrences and program transformations to compute moments in probabilistic loops without external invariants.

Findings

Method effectively computes higher moments for complex probabilistic loops

Applicable to programs with uncountable state spaces

Demonstrated on challenging real-world examples

Abstract

We present a novel static analysis technique to derive higher moments for program variables for a large class of probabilistic loops with potentially uncountable state spaces. Our approach is fully automatic, meaning it does not rely on externally provided invariants or templates. We employ algebraic techniques based on linear recurrences and introduce program transformations to simplify probabilistic programs while preserving their statistical properties. We develop power reduction techniques to further simplify the polynomial arithmetic of probabilistic programs and define the theory of moment-computable probabilistic loops for which higher moments can precisely be computed. Our work has applications towards recovering probability distributions of random variables and computing tail probabilities. The empirical evaluation of our results demonstrates the applicability of our work on many challenging examples.