Nonparametric Bayesian methods for one-dimensional diffusion models

In this paper we review recently developed methods for nonparametric Bayesian inference for one-dimensional diffusion models. We discuss different possible prior distributions, computational issues, and asymptotic results

Some Notes on the Pantun Storytelling of the Baduy Minority Group Its Written and Audiovisual Documentation

Baduy pantun stories are part of the larger Sundanese oral tradition of pantun storytelling in west Java. The stories recount the deeds of the nobility of such old Sundanese kingdoms as Pajajaran and Galuh. Although the Baduy still recite the pantun stories in their rituals, in the larger cities to the east of the Baduy village Kanékés pantun recitation almost disappeared. On the basis of short periods of fieldwork in and around Kanékés village between 1976 and 2014, in this essay I shall discuss Baduy pantun storytelling. I shall summarize earlier major publications and analyse some performance aspects of two Baduy pantun stories which I recorded. Although I do not concentrate on the text, I do discuss a few cultural issues arising from the texts. Baduy oral literature also includes children\u27s and women\u27s songs, as well as fables and myths of origin (dongéng) which do not involve music. These will not be discussed here

Optimality of Poisson processes intensity learning with Gaussian processes

In this paper we provide theoretical support for the so-called "Sigmoidal Gaussian Cox Process" approach to learning the intensity of an inhomogeneous Poisson process on a $d$-dimensional domain. This method was proposed by Adams, Murray and MacKay (ICML, 2009), who developed a tractable computational approach and showed in simulation and real data experiments that it can work quite satisfactorily. The results presented in the present paper provide theoretical underpinning of the method. In particular, we show how to tune the priors on the hyper parameters of the model in order for the procedure to automatically adapt to the degree of smoothness of the unknown intensity and to achieve optimal convergence rates

Minimax lower bounds for function estimation on graphs

We study minimax lower bounds for function estimation problems on large graph when the target function is smoothly varying over the graph. We derive minimax rates in the context of regression and classification problems on graphs that satisfy an asymptotic shape assumption and with a smoothness condition on the target function, both formulated in terms of the graph Laplacian

Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion

In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein's work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an explicit reproducing kernel in the frequency domain. We use these results to derive an extension of the classical Paley-Wiener expansion of the ordinary Brownian motion to the fractional case.Comment: Published at http://dx.doi.org/10.1214/009117904000000955 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Gaussian process methods for one-dimensional diffusions: optimal rates and adaptation

We study the performance of nonparametric Bayes procedures for one-dimensional diffusions with periodic drift. We improve existing convergence rate results for Gaussian process (GP) priors with fixed hyper parameters. Moreover, we exhibit several possibilities to achieve adaptation to smoothness. We achieve this by considering hierarchical procedures that involve either a prior on a multiplicative scaling parameter, or a prior on the regularity parameter of the GP

Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling

We study random series priors for estimating a functional parameter (f\in L^2[0,1]). We show that with a series prior with random truncation, Gaussian coefficients, and inverse gamma multiplicative scaling, it is possible to achieve posterior contraction at optimal rates and adaptation to arbitrary degrees of smoothness. We present general results that can be combined with existing rate of contraction results for various nonparametric estimation problems. We give concrete examples for signal estimation in white noise and drift estimation for a one-dimensional SDE

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