The goal of this paper is to demonstrate the general modeling and practical simulation of random equations with mixture model parameter random variables. Random equations, understood as stationary (non-dynamical) equations with parameters as random variables, have a long history and a broad range of applications. The specific novelty of this explorative study lies on the demonstration of the combinatorial complexity of these equations with mixture model parameters. In a Bayesian argumentation framework we derive a general likelihood function and posterior density of approximate best fit solutions while avoiding significant restrictions about the type of nonlinearity of the equation or mixture models, and demonstrate their numerically efficient implementation for the applied researcher. In the results section, we are specifically focusing on expressive example simulations of approximate likelihood/posterior solutions for random linear equation systems, nonlinear systems of random conic section equations, as well as applications to portfolio optimization, stochastic control and random matrix theory in order to show the wide applicability of the presented methodology.

翻译：本文旨在展示具有混合模型参数随机变量的随机方程的通用建模与实用仿真。随机方程被理解为参数为随机变量的静态（非动态）方程，具有悠久的历史和广泛的应用。本探索性研究的具体新颖之处在于展示了这些具有混合模型参数的方程的组合复杂性。在贝叶斯论证框架下，我们推导出近似最佳拟合解的一般似然函数和后验密度，同时避免对方程或混合模型的非线性类型施加显著限制，并展示了面向应用研究者的数值高效实现方法。在结果部分，我们特别聚焦于随机线性方程组、随机圆锥曲线方程组的非线性系统之近似似然/后验解的表达式例仿真，以及该方法在投资组合优化、随机控制和随机矩阵理论中的应用，以展示所提方法的广泛适用性。