4,429 research outputs found
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
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
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 -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
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
Donsker theorems for diffusions: Necessary and sufficient conditions
We consider the empirical process G_t of a one-dimensional diffusion with
finite speed measure, indexed by a collection of functions F. By the central
limit theorem for diffusions, the finite-dimensional distributions of G_t
converge weakly to those of a zero-mean Gaussian random process G. We prove
that the weak convergence G_t\Rightarrow G takes place in \ell^{\infty}(F) if
and only if the limit G exists as a tight, Borel measurable map. The proof
relies on majorizing measure techniques for continuous martingales.
Applications include the weak convergence of the local time density estimator
and the empirical distribution function on the full state space.Comment: Published at http://dx.doi.org/10.1214/009117905000000152 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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