1,200 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
The time-line as a device to enhance recall in standardized research interviews: a split ballot study
A split ballot experiment was performed on a time-line procedure that was designed to increase the accuracy of responses to retrospective questions. The time-line unites several aided recall properties and was applied to help respondents to reconstruct their educational history. The data were collected during two main waves (1987 and 1991) of a longitudinal social survey in the Netherlands ðN 1; 257Þ: In both 1987 and 1991 respondents were asked about their educational history from August 1983 on. The agreement between the 1991 reports and 1987 reports about the period 1983-1987 was used as a measure of recall accuracy. It was hypothesized that the time-line would enhance recall accuracy regarding the number of educational courses attended, the starting year of the courses, and the entire set of types of courses attended. Additionally, it was expected that the time-line would be especially helpful if the difficulty of the recall task was high – that is to say, in the case of a high frequency or low saliency of the courses followed. The general picture of the results is that the time-line procedure improved data quality in most conditions and never resulted in inferior data quality, supporting the assumption that it indeed may enhance recall. Key words: Data collection; recall accuracy; memory cues; life history. 1
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
The Bayes Lepski's Method and Credible Bands through Volume of Tubular Neighborhoods
For a general class of priors based on random series basis expansion, we
develop the Bayes Lepski's method to estimate unknown regression function. In
this approach, the series truncation point is determined based on a stopping
rule that balances the posterior mean bias and the posterior standard
deviation. Equipped with this mechanism, we present a method to construct
adaptive Bayesian credible bands, where this statistical task is reformulated
into a problem in geometry, and the band's radius is computed based on finding
the volume of certain tubular neighborhood embedded on a unit sphere. We
consider two special cases involving B-splines and wavelets, and discuss some
interesting consequences such as the uncertainty principle and self-similarity.
Lastly, we show how to program the Bayes Lepski stopping rule on a computer,
and numerical simulations in conjunction with our theoretical investigations
concur that this is a promising Bayesian uncertainty quantification procedure.Comment: 42 pages, 2 figures, 1 tabl
Empirical processes indexed by estimated functions
We consider the convergence of empirical processes indexed by functions that
depend on an estimated parameter and give several alternative conditions
under which the ``estimated parameter'' can be replaced by its natural
limit uniformly in some other indexing set . In particular we
reconsider some examples treated by Ghoudi and Remillard [Asymptotic Methods in
Probability and Statistics (1998) 171--197, Fields Inst. Commun. 44 (2004)
381--406]. We recast their examples in terms of empirical process theory, and
provide an alternative general view which should be of wide applicability.Comment: Published at http://dx.doi.org/10.1214/074921707000000382 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) 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
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
The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors
We consider the horseshoe estimator due to Carvalho, Polson and Scott (2010)
for the multivariate normal mean model in the situation that the mean vector is
sparse in the nearly black sense. We assume the frequentist framework where the
data is generated according to a fixed mean vector. We show that if the number
of nonzero parameters of the mean vector is known, the horseshoe estimator
attains the minimax risk, possibly up to a multiplicative constant. We
provide conditions under which the horseshoe estimator combined with an
empirical Bayes estimate of the number of nonzero means still yields the
minimax risk. We furthermore prove an upper bound on the rate of contraction of
the posterior distribution around the horseshoe estimator, and a lower bound on
the posterior variance. These bounds indicate that the posterior distribution
of the horseshoe prior may be more informative than that of other one-component
priors, including the Lasso.Comment: This version differs from the final published version in pagination
and typographical detail; Available at
http://projecteuclid.org/euclid.ejs/141813426
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