Adaptive Priors based on Splines with Random Knots

Splines are useful building blocks when constructing priors on nonparametric models indexed by functions. Recently it has been established in the literature that hierarchical priors based on splines with a random number of equally spaced knots and random coefficients in the B-spline basis corresponding to those knots lead, under certain conditions, to adaptive posterior contraction rates, over certain smoothness functional classes. In this paper we extend these results for when the location of the knots is also endowed with a prior. This has already been a common practice in MCMC applications, where the resulting posterior is expected to be more "spatially adaptive", but a theoretical basis in terms of adaptive contraction rates was missing. Under some mild assumptions, we establish a result that provides sufficient conditions for adaptive contraction rates in a range of models

Rate-optimal Bayesian intensity smoothing for inhomogeneous Poisson processes

We apply nonparametric Bayesian methods to study the problem of estimating the intensity function of an inhomogeneous Poisson process. We exhibit a prior on intensities which both leads to a computationally feasible method and enjoys desirable theoretical optimality properties. The prior we use is based on B-spline expansions with free knots, adapted from well-established methods used in regression, for instance. We illustrate its practical use in the Poisson process setting by analyzing count data coming from a call centre. Theoretically we derive a new general theorem on contraction rates for posteriors in the setting of intensity function estimation. Practical choices that have to be made in the construction of our concrete prior, such as choosing the priors on the number and the locations of the spline knots, are based on these theoretical findings. The results assert that when properly constructed, our approach yields a rate-optimal procedure that automatically adapts to the regularity of the unknown intensity function

Efficient Estimation of Analytic Density under Random Censorship

this paper we establish, under condition that the censoring is not too severe, the exact limiting behavior of the local minimax risk up to a constant and show that the estimator of the form (1), with a properly chosen kernel, is locally asymptotically efficient. We emphasize here that the choice of nonparametric class (analytic densities) has made it possible. We propose a wide class of kernels on which the estimator can be based, which turns out to be important in the estimation problem with censored observations. Using the martingale approach enables us to derive the exact upper bound for the local minimax risk. The lower bound for the local minimax risk is based on the elementary van Trees inequality (Gill and Levit (1995)). 2 Definitions and main result

Efficient estimation of analytic density under random censorship

The nonparametric minimax estimation of an analytic density at a given point, under random censorship, is considered. Although the problem of estimating density is known to be irregular in a certain sense, we make some connections relating this problem to the problem of estimating smooth functionals. Under condition that the censoring is not too severe, we establish the exact limiting behaviour of the local minimax risk and propose the efficient (locally asymptotically minimax) estimator - an integral of some kernel with respect to the Kaplan-Meier estimator