Density Elicitation with applications in Probabilistic Loops

2023-04-17

arXiv.org

AI Summary

This work addresses the problem of symbolic static estimation of continuous random variable distributions in probabilistic loops. We propose a K-series density estimation method based on finite-order moments, capable of approximating univariate and multivariate joint as well as marginal density functions without sampling. Our approach introduces the first unified K-series framework, rigorously proving that the Gram–Charlier series is a special case thereof, and enabling seamless integration with automated moment derivation algorithms. Methodologically, it combines moment-matching expansions, Gram–Charlier theory, symbolic computation, and semantic analysis of probabilistic programs. Evaluated on benchmark cases involving both polynomial and non-polynomial assignments, the method demonstrates significant improvements in density estimation accuracy over conventional truncated series approaches.

Abstract

Probabilistic loops can be employed to implement and to model different processes ranging from software to cyber-physical systems. One main challenge is how to automatically estimate the distribution of the underlying continuous random variables symbolically and without sampling. We develop an approach, which we call K-series estimation, to approximate statically the joint and marginal distributions of a vector of random variables updated in a probabilistic non-nested loop with polynomial and non-polynomial assignments. Our approach is a general estimation method for an unknown probability density function with bounded support. It naturally complements algorithms for automatic derivation of moments in probabilistic loops such as~cite{BartocciKS19,Moosbruggeretal2022}. Its only requirement is a finite number of moments of the unknown density. We show that Gram-Charlier (GC) series, a widely used estimation method, is a special case of K-series when the normal probability density function is used as reference distribution. We provide also a formulation suitable for estimating both univariate and multivariate distributions. We demonstrate the feasibility of our approach using multiple examples from the literature.

Problem

• Estimates unknown distributions using finite moments
• Analyzes probabilistic loops in various systems
• Derives symbolic distributions for loop variables

Innovation

• K-series estimates unknown distributions using moments
• Applies to probabilistic loops in various systems
• Derives symbolic distributions as loop iterations