Many stochastic continuous-state dynamical systems can be modeled as probabilistic programs with nonlinear non-polynomial updates in non-nested loops. We present two methods, one approximate and one exact, to automatically compute, without sampling, moment-based invariants for such probabilistic programs as closed-form solutions parameterized by the loop iteration. The exact method applies to probabilistic programs with trigonometric and exponential updates and is embedded in the Polar tool. The approximate method for moment computation applies to any nonlinear random function as it exploits the theory of polynomial chaos expansion to approximate non-polynomial updates as the sum of orthogonal polynomials. This translates the dynamical system to a non-nested loop with polynomial updates, and thus renders it conformable with the Polar tool that computes the moments of any order of the state variables. We evaluate our methods on an extensive number of examples ranging from modeling monetary policy to several physical motion systems in uncertain environments. The experimental results demonstrate the advantages of our approach with respect to the current state-of-the-art.

翻译：许多随机连续状态动力系统可以建模为包含非线性非多项式更新的非嵌套循环概率程序。我们提出了两种方法——一种近似方法和一种精确方法——用于在不进行采样的情况下，自动计算此类概率程序的矩不变量，并将其表示为循环迭代参数的闭式解。精确方法适用于包含三角函数和指数更新的概率程序，并已集成于Polar工具中。近似矩计算方法适用于任意非线性随机函数，它利用多项式混沌展开理论，将非多项式更新近似为正交多项式之和。这将动力系统转化为具有多项式更新的非嵌套循环，从而使其与可计算状态变量任意阶矩的Polar工具兼容。我们通过从货币政策建模到不确定环境下多种物理运动系统的大量示例评估了所提方法。实验结果表明，我们的方法相较于当前最先进技术具有显著优势。