2,542 research outputs found
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
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
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