831 research outputs found
Rates of contraction of posterior distributions based on Gaussian process priors
We derive rates of contraction of posterior distributions on nonparametric or
semiparametric models based on Gaussian processes. The rate of contraction is
shown to depend on the position of the true parameter relative to the
reproducing kernel Hilbert space of the Gaussian process and the small ball
probabilities of the Gaussian process. We determine these quantities for a
range of examples of Gaussian priors and in several statistical settings. For
instance, we consider the rate of contraction of the posterior distribution
based on sampling from a smooth density model when the prior models the log
density as a (fractionally integrated) Brownian motion. We also consider
regression with Gaussian errors and smooth classification under a logistic or
probit link function combined with various priors.Comment: Published in at http://dx.doi.org/10.1214/009053607000000613 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
We consider nonparametric Bayesian estimation inference using a rescaled
smooth Gaussian field as a prior for a multidimensional function. The rescaling
is achieved using a Gamma variable and the procedure can be viewed as choosing
an inverse Gamma bandwidth. The procedure is studied from a frequentist
perspective in three statistical settings involving replicated observations
(density estimation, regression and classification). We prove that the
resulting posterior distribution shrinks to the distribution that generates the
data at a speed which is minimax-optimal up to a logarithmic factor, whatever
the regularity level of the data-generating distribution. Thus the hierachical
Bayesian procedure, with a fixed prior, is shown to be fully adaptive.Comment: Published in at http://dx.doi.org/10.1214/08-AOS678 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Frequentist coverage of adaptive nonparametric Bayesian credible sets
We investigate the frequentist coverage of Bayesian credible sets in a
nonparametric setting. We consider a scale of priors of varying regularity and
choose the regularity by an empirical Bayes method. Next we consider a central
set of prescribed posterior probability in the posterior distribution of the
chosen regularity. We show that such an adaptive Bayes credible set gives
correct uncertainty quantification of "polished tail" parameters, in the sense
of high probability of coverage of such parameters. On the negative side, we
show by theory and example that adaptation of the prior necessarily leads to
gross and haphazard uncertainty quantification for some true parameters that
are still within the hyperrectangle regularity scale.Comment: Published at http://dx.doi.org/10.1214/14-AOS1270 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Rejoinder to discussions of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"
Rejoinder of "Frequentist coverage of adaptive nonparametric Bayesian
credible sets" by Szab\'o, van der Vaart and van Zanten [arXiv:1310.4489v5].Comment: Published at http://dx.doi.org/10.1214/15-AOS1270REJ in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Bayesian inverse problems with Gaussian priors
The posterior distribution in a nonparametric inverse problem is shown to
contract to the true parameter at a rate that depends on the smoothness of the
parameter, and the smoothness and scale of the prior. Correct combinations of
these characteristics lead to the minimax rate. The frequentist coverage of
credible sets is shown to depend on the combination of prior and true
parameter, with smoother priors leading to zero coverage and rougher priors to
conservative coverage. In the latter case credible sets are of the correct
order of magnitude. The results are numerically illustrated by the problem of
recovering a function from observation of a noisy version of its primitive.Comment: Published in at http://dx.doi.org/10.1214/11-AOS920 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Small Deviations of Smooth Stationary Gaussian Processes
We investigate the small deviation probabilities of a class of very smooth
stationary Gaussian processes playing an important role in Bayesian statistical
inference. Our calculations are based on the appropriate modification of the
entropy method due to Kuelbs, Li, and Linde as well as on classical results
about the entropy of classes of analytic functions. They also involve
Tsirelson's upper bound for small deviations and shed some light on the limits
of sharpness for that estimate
Bayesian recovery of the initial condition for the heat equation
We study a Bayesian approach to recovering the initial condition for the heat
equation from noisy observations of the solution at a later time. We consider a
class of prior distributions indexed by a parameter quantifying "smoothness"
and show that the corresponding posterior distributions contract around the
true parameter at a rate that depends on the smoothness of the true initial
condition and the smoothness and scale of the prior. Correct combinations of
these characteristics lead to the optimal minimax rate. One type of priors
leads to a rate-adaptive Bayesian procedure. The frequentist coverage of
credible sets is shown to depend on the combination of the prior and true
parameter as well, with smoother priors leading to zero coverage and rougher
priors to (extremely) conservative results. In the latter case credible sets
are much larger than frequentist confidence sets, in that the ratio of
diameters diverges to infinity. The results are numerically illustrated by a
simulated data example.Comment: 17 pages, 4 figures. Published in Comm. Statist. Theory Methods. This
version differs from the original in pagination and typographic detail. arXiv
admin note: text overlap with arXiv:1103.269
UvA-DARE (Digital Academic Repository) Empirical Bayes scaling of Gaussian priors in the white noise model
The association between medical education accreditation and the examination performance of internationally educated physicians seeking certification in the United States
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