Bayesian curve estimation on bounded intervals

Bayesian methods of curve estimation are becoming increasingly popular with increasing computational abilities and better understanding of their theoretical properties. For instance, Bayesian density estimation on the line with a prior a Dirichlet mixture of normals, has been widely used. For a curve such as a probability density on a compact interval, mixtures of betas are however more appropriate. In fact, only a relatively few betas, given by the Bernstein polynomials, can approximate any continuous density on a compact interval. This led to the development of priors based on Bernstein polynomials. The resulting Bayesian density estimates are very sensible and consistent estimates are obtained.

In this talk, we extend the idea of a Bernstein polynomial prior to some other types of curve estimation. Examples include the spectral density of a stationary time series and the response curve of dose levels of a drug. For the spectral density, the observations are dependent. We decorrelate the data by the spectral transform and use the Whittle likelihood to obtain the posterior. In the dose response problem, logistic regression is traditionally used in the literature. However, particularly at toxic levels, a logistic link is inappropriate and monotonicity of the response curve seems to be also questionable. A completely flexible nonparametric model for the dose response curve is therefore of substantial interest. We show that, with appropriate modifications, Bernstein polynomial priors can be constructed for these curves. We also consider a Gaussian process type prior for the dose response curve. The resulting posteriors are amenable to the Markov chain Monte Carlo method of computation and lead to sensible estimates. We show that the posterior distributions are consistent in appropriate distances.