Optimal Rates of Convergence to Bayes Risk in Nonparametric Discrimination

Optimal Rates of Convergence to Bayes Risk in Nonparametric Discrimination

On the Rates of Convergence from Surrogate Risk Minimizers to the Bayes Optimal Classifier

We study the rates of convergence from empirical surrogate risk minimizers to the Bayes optimal classifier. Specifically, we introduce the notion of consistency intensity to characterize a surrogate loss function and exploit this notion to obtain the rate of convergence from an empirical surrogate risk minimizer to the Bayes optimal classifier, enabling fair comparisons of the excess risks of d...

Nonparametric Bayes-risk estimation

Absrract-Two nonparametric methods to estimate the Bayes risk using classified sample sets are described and compared. The first method uses the nearest neighbor error rate as an estimate to bound the Bayes risk. The second method estimates the Bayes decision regions by applying Parzen probability-density function estimates and counts errors made using these regions. This estimate is shown to b...

Convergence Rates in Nonparametric Bayesian Density Estimation

We consider Bayesian density estimation for compactly supported densities using Bernstein mixtures of beta-densities equipped with a Dirichlet prior on the distribution function. We derive the rate of convergence for α-smooth densities for 0 < α ≤ 2 and show that a faster rate of convergence can be obtained by using fewer terms in the mixtures than proposed before. The Bayesian procedure adapts...

Convergence Rates of Nonparametric Posterior Distributions

We study the asymptotic behavior of posterior distributions. We present general posterior convergence rate theorems, which extend several results on posterior convergence rates provided by Ghosal and Van der Vaart (2000), Shen and Wasserman (2001) and Walker, Lijor and Prunster (2007). Our main tools are the Hausdorff α-entropy introduced by Xing and Ranneby (2008) and a new notion of prior con...