Triply fuzzy function approximation for hierarchical Bayesian inference

We prove that three independent fuzzy systems can uniformly approximate Bayesian posterior probability density functions by approximating the prior and likelihood probability densities as well as the hyperprior probability densities that underly the 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 approximation 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. Iterative fuzzy Bayesian inference can lead to rule explosion. We prove that conjugacy in the if-part set functions for prior, hyperprior, and likelihood fuzzy approximators reduces rule explosion. We also prove that a type of semi-conjugacy of if-part set functions for those fuzzy approximators results in fewer parameters in the fuzzy posterior approximator.