Distribution Estimation for Probabilistic Loops

TL;DR

This paper introduces an algorithmic method to estimate the distributions of random variables in probabilistic loops using known moments and statistical techniques, validated through experiments on models from finance and biology.

Contribution

It presents a novel approach combining Maximum Entropy and Gram-Charlier series methods for distribution estimation in probabilistic loops, utilizing symbolic and sampling techniques for moments.

Findings

Accurately estimates distributions using moments and statistical tests.

Effective on models with polynomial updates and common distributions.

Validated through experiments on financial and biological models.

Abstract

We present an algorithmic approach to estimate the value distributions of random variables of probabilistic loops whose statistical moments are (partially) known. Based on these moments, we apply two statistical methods, Maximum Entropy and Gram-Charlier series, to estimate the distributions of the loop's random variables. We measure the accuracy of our distribution estimation by comparing the resulting distributions using exact and estimated moments of the probabilistic loop, and performing statistical tests. We evaluate our method on several probabilistic loops with polynomial updates over random variables drawing from common probability distributions, including examples implementing financial and biological models. For this, we leverage symbolic approaches to compute exact higher-order moments of loops as well as use sampling-based techniques to estimate moments from loop executions. Our experimental results provide practical evidence of the accuracy of our method for estimating distributions of probabilistic loop outputs.