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.
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