Moment-based Density Elicitation with Applications in Probabilistic Loops

TL;DR

Moment-based Density Elicitation with K-series introduces a density estimator that reconstructs an unknown pdf $f$ from a finite set of moments using a reference distribution $\phi$ and an orthogonal basis $\{h_i\}$. The method unifies and extends existing series approaches by proving that Gram-Charlier and Method of Moments are special cases, and it provides convergence guarantees in $L_{1}$ together with a moment-matching property. It is particularly suited to probabilistic loops and prob-solvable loops, enabling symbolic, iteration-dependent pdfs and scalable multivariate extensions. Empirical results across robotics, economics, and biology demonstrate high accuracy for both marginal and joint densities and show K-series outperforms nonparametric approaches when moments are available, offering a fast, robust alternative for distribution recovery in uncertain dynamic systems.

Abstract

We propose the K-series estimation approach for the recovery of unknown univariate and multivariate distributions given knowledge of a finite number of their moments. Our method is directly applicable to the probabilistic analysis of systems that can be represented as probabilistic loops; i.e., algorithms that express and implement non-deterministic processes ranging from robotics to macroeconomics and biology to software and cyber-physical systems. K-series statically approximates the joint and marginal distributions of a vector of continuous random variables updated in a probabilistic non-nested loop with nonlinear assignments given a finite number of moments of the unknown density. Moreover, K-series automatically derives the distribution of the systems' random variables symbolically as a function of the loop iteration. K-series density estimates are accurate, easy and fast to compute. We demonstrate the feasibility and performance of our approach on multiple benchmark examples from the literature.

Figures (22)

• Figure 1: Probabilistic loop with non-polynomial assignment for the Differential-Drive Mobile Robot Jasouretal2021 (top left), the approximations of the marginal distributions with K-series (top right), the approximation of the joint distribution with K-series (bottom left) and comparison with true histogram (bottom right).
• Figure 2: K-series approximation of a truncated exponential distribution (panel (a)) and the Irwin-Hall distribution (panel (b)).
• Figure 3: (A) Probabilistic loop with non-polynomial assignment, (B) Transformation of the program A using Polynomial Chaos Expansion Kofnovetal2022, by replacing the function $\min(\cdot, \cdot)$ with the polynomial $G(x,y)$.
• Figure 4: Left panel: First four moments expressed symbolically in the number of iterations. Right panel: Comparison between the histogram of the sampling pdf and the symbolic K-series estimation at $t=30$.
• Figure 5: Turning Vehicle Model: Code and K-series estimates of the marginal pdfs of $X$ and $Y$ (right upper and lower panels), the joint (lower left panel) and comparison bar plot (upper left panel) at iteration $t = 20$.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3: Moment matching
• proof
• Theorem 4
• Theorem 5
• Theorem 6