Uniform Convergence of Probability Mixtures that Represent Combined Fuzzy Systems

We can combine expert knowledge by combining the probability mixtures that represent the if-then rules of the experts. Fuzzy rules define a generalized probability mixture whose moments describe a fuzzy system and its uncertainty. The mixture’s Bayesian structure gives a complete posterior probability description of the if-then fuzzy-set rules as they fire. A new theorem extends the uniform convergence of a fuzzy system’s mixture to the uniform convergence of the sequence of expert mixtures that represent any number of combined fuzzy systems as they each converge to a target function. A mixture of just two normal bell curves exactly represents the target function in the scalar case and serves as the probabilistic target of the converging mixture sequence. A sampled deep neural network can serve as the target function. Then the mixture defines a proxy system that gives a probabilistic form of explainable AI. The uniform convergence result extends to any continuous transformation of the converging fuzzy systems and further extends to the uniform mixture convergence of any continuous function of the combined systems and their continuous transformations.