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：在一般框架下探究非线性方程组的解具有挑战性，尤其当方程中包含由概率密度参数化的不确定性参数时。此类随机方程（即参数为随机变量的稳态非动力方程）具有悠久历史及广泛应用领域。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可能性。在避免对非线性类型或混合模型施加重大限制的前提下，我们推导出近似最优拟合解的一般似然函数和后验密度，并为应用研究者展示其数值高效实现方案。结果部分重点针对以下案例进行模拟分析：随机线性方程组的近似似然/后验解、随机圆锥曲线非线性方程组，以及在投资组合优化、随机控制与随机矩阵理论中的应用，以充分论证所提出方法的广泛适用性。