Adaptive Bayesian Procedures Using Random Series Priors

By Weining Shen, Subhashis Ghosal. 2014

93 citations recorded by the source corpus.

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/141bea23b40ad7c754c803409da7be003783f881

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Citations in the corpus

Using Stacking to Average Bayesian Predictive Distributions (with Discussion) (2017, background): “…have appealing frequentist asymptotic properties, such as appropriate notions of consistency (Schwartz, 1965) and optimal rates of convergence (van der Vaart and van Zanten, 2009; Bhattacharya et al., 2014; Castillo, 2014; Shen and Ghosal, 2015; Li and Ghosal, 2017; Ghosal and van der Vaart, 2017).”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, methodology): “Shen and Ghosal [28, 29] used tensor products of B-splines, respectively, for Bayesian multivariate density estimation and high dimensional density regression in the anisotropic case.”
Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator (2015, methodology): “In the hierarchical setup with a prior on k, type (T1) prior has been studied by [1, 25] for generic models, by [22] for density estimation, by [2] for Gaussian white noise model and by [20] for inverse problems.”