Investigating solutions of nonlinear equation systems is challenging in a general framework, especially if the equations contain uncertainties about parameters modeled by probability densities. Such random equations, understood as stationary (non-dynamical) equations with parameters as random variables, have a long history and a broad range of applications. In this work, we study nonlinear random equations by combining them with mixture model parameter random variables in order to investigate the combinatorial complexity of such equations and how this can be utilized practically. We derive a general likelihood function and posterior density of approximate best fit solutions while avoiding significant restrictions about the type of nonlinearity or mixture models, and demonstrate their numerically efficient application for the applied researcher. In the results section we are specifically focusing on 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：在一般框架下研究非线性方程组的解具有挑战性，尤其是当方程包含由概率密度建模的参数不确定性时。此类随机方程——理解为以随机变量为参数的静态（非动态）方程——具有悠久的历史和广泛的应用。本研究通过将非线性随机方程与混合模型参数随机变量相结合，探讨此类方程的组合复杂性及其实际应用方式。我们推导了近似最优拟合解的一般似然函数和后验密度，同时避免对非线性类型或混合模型施加显著限制，并为应用研究者展示了其数值高效的应用方法。在结果部分，我们重点展示了随机线性方程组、随机圆锥曲线非线性方程组的近似似然/后验解示例模拟，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案例，以证明所提方法具有广泛的适用性。