Computational Framework for Bézier Distributions

2026-06-22

AI Summary

Existing methods lack the capability to efficiently fit flexible continuous univariate distributions with bounded support—such as Bézier distributions—thereby limiting their applicability in stochastic simulation and decision analysis. This work proposes a first-order optimization-based computational framework that leverages maximum likelihood estimation and minimum error criteria to fit Bézier distributions efficiently. The key innovation lies in constructing an asymptotically lossless convex feasible set and designing an efficient projection operator based on isotonic regression, which substantially enhances both computational efficiency and numerical robustness. The authors release an open-source Python package, bezierv, offering a unified interface that achieves stable and reliable fitting on real-world data. Empirical results demonstrate speedups of three to four orders of magnitude over traditional derivative-free methods and approximately three orders of magnitude over IPOPT, while maintaining comparable accuracy and exhibiting superior robustness.

Abstract

Flexible continuous univariate distributions with bounded support are essential for accurate input modeling in stochastic simulation and decision analysis. Although Bézier distributions provide a powerful family capable of representing complex shapes, their adoption has been hindered by the lack of efficient fitting procedures and modern software implementations. This paper develops a computational framework for fitting Bézier distributions to empirical data via both minimum error and maximum likelihood estimation, leveraging first-order optimization methods and exploiting the geometry of the parameter space. We identify provably (asymptotically) lossless convex restrictions of the feasible set that enable efficient projection operators based on isotonic regression and develop first-order algorithms that reduce computational runtime by three to four orders of magnitude compared to traditional derivative-free methods, while delivering consistent fits across real-world data. When benchmarked against the nonlinear solver IPOPT, our methods prove three orders of magnitude faster on average and more robust, while achieving comparable accuracy. To bridge the gap between theory and practice, we introduce bezierv, an open-source Python package providing a unified interface for fitting, analyzing, and convolving Bézier distributions.

Problem

• Bézier distributions
• input modeling
• efficient fitting
• stochastic simulation
• bounded support

Innovation

• Bézier distributions
• first-order optimization
• isotonic regression
• convex restriction
• stochastic modeling