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

Recursive estimation of a drifted autoregressive parameter

Suppose the X0,...., Xn are observations of a one-dimensional stochastic dynamic process described by autoregression equations when the autoregressive parameter is drifted with time, i.e. it is some function of time: ¿0,...., ¿n, with ¿k = ¿ (k/n). The function ¿(t) is assumed to belong a priori to a predetermined nonparametric class of functions satisfying the Lipschitz smoothness condition. At each time point t those observations are accessible which have been obtained during the preceding time interval. A recursive algorithm is proposed to estimate ¿??t??. Under some conditions on the model, we derive the rate of convergence of the proposed estimator when the frequencyof observations n tends to infinity

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

Adaptive Bayesian inference on the mean of an infinite-dimensional normal distribution

We consider the problem of estimating the mean of an infinite-break dimensional normal distribution from the Bayesian perspective. Under the assumption that the unknown true mean satisfies a smoothness condition, we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators. Although the posterior distribution can achieve the optimal rate of convergence, the required prior depends on a smoothness parameter q. When this parameter q is unknown, besides the estimation of the mean, we encounter the problem of selecting a model. In a Bayesian approach, this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model. We show that if q takes values only in a discrete set, the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model. A slightly weaker result is presented when q is unrestricted. An adaptive point estimator based on the posterior distribution is also constructed. Primary Subjects: 62G20. Secondary Subjects: 62C10, 62G05. Keywords: Adaptive Bayes procedure; convergence rate; minimax risk; posterior distribution; model selection

Asymptotically local minimax estimation of infinitely smooth density with censored data

The problem of the nonparametric minimax estimation of an innitely smooth density at a given point, under random censorship, is considered. We establish the exact limiting behavior of the local minimax risk and propose the ecient kernel -type estimator based on the Kaplan-Meier estimator