  We consider priors for several nonparametric Bayesian models which use finite
random series with a random number of terms. The prior is constructed through
distributions on the number of basis functions and the associated coefficients.
We derive a general result on the construction of an appropriate sieve and
obtain adaptive posterior contraction rates for all smoothness levels of the
function in the true model. We apply this general result on several statistical
problems such as signal processing, density estimation, nonparametric additive
regression, classification, spectral density estimation, functional regression
etc. The prior can be viewed as an alternative to commonly used Gaussian
process prior, but can be analyzed by relatively simpler techniques and in many
cases allows a simpler approach to computation without using Markov chain
Monte-Carlo (MCMC) methods. A simulation study was conducted to show that the
performance of the random series prior is comparable to that of a Gaussian
process prior.
