Triply Fuzzy Function Approximation for Bayesian Inference

We prove that independent fuzzy systems can uniformly approximate Bayesian posterior probability density functions by approximating prior and likelihood probability densities as well as hyperprior probability densities that underly priors. This triply fuzzy function approximation extends the recent theorem for uniformly approximating the posterior density by approximating just the prior and likelihood densities. This allows users to state priors and hyper-priors in words or rules as well as to adapt them from sample data. A fuzzy system with just two rules can exactly represent common closed-form probability densities so long as they are bounded. The function approximators can also be neural networks or any other type of uniform function approximator.