We consider a prior for nonparametric Bayesian estimation which uses finiterandom series with a random number of terms. The prior is constructed throughdistributions on the number of basis functions and the associated coefficients.We derive a general result on adaptive posterior convergence rates for allsmoothness levels of the function in the true model by constructing anappropriate "sieve" and applying the general theory of posterior convergencerates. We apply this general result on several statistical problems such assignal processing, density estimation, various nonparametric regressions,classification, spectral density estimation, functional regression etc. Theprior can be viewed as an alternative to the commonly used Gaussian processprior, but properties of the posterior distribution can be analyzed byrelatively simpler techniques and in many cases allows a simpler approach tocomputation without using Markov chain Monte-Carlo (MCMC) methods. A simulationstudy is conducted to show that the accuracy of the Bayesian estimators basedon the random series prior and the Gaussian process prior are comparable. Weapply the method on two interesting data sets on functional regression.
展开▼
