Convergence rates for Bayesian density estimation of infinite-dimensional exponential families
By Catia Scricciolo. 2006
51 citations recorded by the source corpus.
We study the rate of convergence of posterior distributions in density estimation problems for log-densities in periodic Sobolev classes characterized by a smoothness parameter p. The posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under Hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes. A prior on the density class of interest is induced by a prior on the coefficients of the trigonometric series expansion of the log-density. We show that when p is known, the posterior distribution of a Gaussian prior achieves the optimal rate provided the prior variances die off sufficiently rapidly. For a mixture of normal distributions, the mixing weights on the dimension of the exponential family are assumed to be bounded below by an exponentially decreasing sequence. To avoid the use of infinite bases, we develop priors that cut off the series at a sample-size-dependent truncation point. When the degree of smoothness is unknown, a finite mixture of normal priors indexed by the smoothness parameter, which is also assigned a prior, produces the best rate. A rate-adaptive estimator is derived.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/45d89f5b35cb0ceffcc8e8c36c326256e3eee2dc
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In Artificial Intelligence
- Judith Rousseau
- Subhashis Ghosal
- Weining Shen
- Vincent Rivoirard
- I. Castillo
- Richard Nickl
- Kolyan Ray
- A. van der Vaart
In Statistics and Probability
- Judith Rousseau
- Subhashis Ghosal
- Weining Shen
- I. Castillo
- Vincent Rivoirard
- Richard Nickl
- Kolyan Ray
- A. van der Vaart
Citations in the corpus
- Posterior concentration rates for infinite dimensional exponential families (2012, background): “Moreover Scricciolo [18] merely considers Gaussian priors.”
- Adaptive Bayesian Procedures Using Random Series Priors (2014, methodology): “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.”
- BERNSTEIN-VON MISES THEOREM FOR LINEAR FUNCTIONALS OF THE DENSITY (2009, background): “So, we dene a prior on the set F = [kFk by dening a prior N and then, once k is chosen, we x a prior k on Fk. Such priors are often considered in the Bayesian non parametric literature. See for instance [ 19 ].”