Combinatorial Potential of Random Equations with Mixture Models: Modeling and Simulation

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.