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.
翻译:概率循环可用于实现和建模从软件到物理系统等不同的过程。其主要挑战之一是如何在不进行抽样的情况下符号地自动估计潜在连续随机变量的分布。我们开发了一种称为K系列估计的方法,用于静态近似地估计在具有多项式和非多项式赋值的概率非嵌套循环中更新的随机变量向量的联合和边际分布。我们的方法是未知概率密度函数的通用估计方法,具有有界支持。它自然地补充了概率循环中自动导出时刻的算法,例如~\cite{BartocciKS19,Moosbruggeretal2022}。它唯一的要求是未知密度的有限时刻。我们表明了Gram-Charlier(GC)级数,一种广泛使用的估计方法,是当使用正态概率密度函数作为参考分布时的一种特殊情况。我们还提供了适用于估计单变量和多变量分布的公式。我们使用文献中的多个示例演示了我们方法的可行性。