# Random Bernstein Polynomials

@article{Petrone1999RandomBP, title={Random Bernstein Polynomials}, author={Sonia Petrone}, journal={Scandinavian Journal of Statistics}, year={1999}, volume={26} }

Random Bernstein polynomials which are also probability distribution functions on the closed unit interval are studied. The probability law of a Bernstein polynomial so defined 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…

## 163 Citations

### Consistency of Bernstein polynomial posteriors

- Mathematics
- 2002

A Bernstein prior is a probability measure on the space of all the distribution functions on [0, 1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose…

### Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

- Mathematics, Computer ScienceComput. Stat. Data Anal.
- 2015

### The Random Bernstein Polynomial Smoothing Via ABC Method

- Mathematics
- 2017

In recent years, many statistical inference problems have been solved by using Markov Chain Monte Carlo (MCMC) techniques. However, it is necessary to derivate the analytical form for the likelihood…

### Convergence rates for density estimation with Bernstein polynomials

- Mathematics
- 2001

Mixture models for density estimation provide a very useful set up for the Bayesian or the maximum likelihood approach. For a density on the unit interval, mixtures of beta densities form a flexible…

### Feller operators and mixture priors in Bayesian nonparametrics

- Mathematics, Computer Science
- 2010

It is proved that, when the random elements used in their construction are chosen in the natural exponential family, they have several properties of interest in statistical applications, and can be represented as mixtures of simple probability distribution functions.

### Efficient and robust density estimation using Bernstein type polynomials

- Mathematics
- 2014

A method of parameterising and smoothing the unknown underlying distributions using Bernstein type polynomials with positive coefficients is proposed, verified and investigated. Any distribution with…

### Bayesian density estimation using bernstein polynomials

- Computer Science, Mathematics
- 1999

A Bayesian nonparametric procedure for density estimation, for data in a closed, bounded interval, say [0,1], using a prior based on Bemstein polynomials to express the density as a mixture of given beta densities, with random weights and a random number of components.

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