Random Bernstein Polynomials

Abstract

Random Bernstein polynomials which are also probability distribution functions on the closed unit interval are studied. The probability law of a Bernstein polynomial so de®ned provides a novel prior on the space of distribution functions on [0, 1], which has full support and can easily select absolutely continuous distribution functions with a continuous and smooth derivative. In particular, the Bernstein polynomial which approximates a Dirichlet process is studied. This may be of interest in Bayesian non-parametric inference. In the second part of the paper, we study the posterior from a``Bernstein±Dirichlet'' prior and suggest a hybrid Monte Carlo approximation of it. The proposed algorithm has some aspects of novelty since the problem under examination has a``changing dimension'' parameter space.

Bayesian nonparametric inference of stochastically ordered distributions, with Pólya trees and Bernstein polynomials

“…In this paper we show how a Bayesian nonparametric approach can be implemented in a simple way, using either Pólya tree prior distributions (Ferguson, 1974), or Bernstein polynomial prior distributions (Petrone, 1999a(Petrone, , 1999b. It is also well known that both types support to absolutely-continuous distributions.…”

“…The Bernstein polynomial prior distribution, introduced by Petrone (1999aPetrone ( , 1999b, assumes that a univariate random density function f (with sample space domain [0; 1]) has the representation:…”

Bayesian Methods for Function Estimation

“…Using an idea of Diaconis that this approximation property may be exploited to construct priors with full topological support, Petrone (1999aPetrone ( , 1999b proposed the following hierarchical prior called the Bernstein polynomial prior: Petrone (1999a) showed that if for all k, ρ(k) > 0 and w k has full support on ∆ k , then every distribution on (0, 1] is in the weak support of the Bernstein polynomial prior, and every continuous distribution is in the topological support of the prior defined by the Kolmogorov-Smirnov distance.…”

“…, x n ). Petrone (1999aPetrone ( , 1999b discussed MCMC algorithms to compute the posterior expectations and carried out extensive simulations to show that the resulting density estimates work well.…”

On the analyticity of functions approximated by their q-Bernstein polynomials when q>1

“…The importance of the Bernstein polynomials opened the gates to the discovery of their numerous generalizations as well as their applications in various mathematical disciplines, see, for example, [1][2][3][4][5][6][7][8]. Due to the speedy development of the qcalculus, recent generalizations based on the q-integers have emerged.…”