The goal of this paper is to demonstrate the general modeling and practical simulation of approximate solutions 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 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 showcasing the combinatorial potential of random linear equation systems and nonlinear systems of random conic section equations. Introductory applications to portfolio optimization, stochastic control and random matrix theory are provided 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近似解的通用建模与实用仿真。随机方程——即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创新点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，我们推导了近似最优拟合解的似然函数和后验密度，同时避免了对方程非线性类型或混合模型的显著限制，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，我们重点通过具有表现力的示例仿真，展示了随机线性方程组和随机圆锥曲线非线性方程组的组合潜力。文中还提供了在投资组合优化、随机控制和随机矩阵理论中的初步应用，以展示所提方法的广泛适用性。