1,092 research outputs found
On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures
We continue the investigation of Bernstein-von Mises theorems for
nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999-2028]. We
introduce multiscale spaces on which nonparametric priors and posteriors are
naturally defined, and prove Bernstein-von Mises theorems for a variety of
priors in the setting of Gaussian nonparametric regression and in the i.i.d.
sampling model. From these results we deduce several applications where
posterior-based inference coincides with efficient frequentist procedures,
including Donsker- and Kolmogorov-Smirnov theorems for the random posterior
cumulative distribution functions. We also show that multiscale posterior
credible bands for the regression or density function are optimal frequentist
confidence bands.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1246 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Large sample asymptotics for the two-parameter Poisson--Dirichlet process
This paper explores large sample properties of the two-parameter
Poisson--Dirichlet Process in two contexts. In a Bayesian
context of estimating an unknown probability measure, viewing this process as a
natural extension of the Dirichlet process, we explore the consistency and weak
convergence of the the two-parameter Poisson--Dirichlet posterior process. We
also establish the weak convergence of properly centered two-parameter
Poisson--Dirichlet processes for large This latter result
complements large results for the Dirichlet process and
Poisson--Dirichlet sequences, and complements a recent result on large
deviation principles for the two-parameter Poisson--Dirichlet process. A
crucial component of our results is the use of distributional identities that
may be useful in other contexts.Comment: Published in at http://dx.doi.org/10.1214/074921708000000147 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions
The problem of determining a periodic Lipschitz vector field from an observed trajectory of the solution of the
multi-dimensional stochastic differential equation \begin{equation*} dX_t =
b(X_t)dt + dW_t, \quad t \geq 0, \end{equation*} where is a standard
-dimensional Brownian motion, is considered. Convergence rates of a
penalised least squares estimator, which equals the maximum a posteriori (MAP)
estimate corresponding to a high-dimensional Gaussian product prior, are
derived. These results are deduced from corresponding contraction rates for the
associated posterior distributions. The rates obtained are optimal up to
log-factors in -loss in any dimension, and also for supremum norm loss
when . Further, when , nonparametric Bernstein-von Mises
theorems are proved for the posterior distributions of . From this we deduce
functional central limit theorems for the implied estimators of the invariant
measure . The limiting Gaussian process distributions have a covariance
structure that is asymptotically optimal from an information-theoretic point of
view.Comment: 55 pages, to appear in the Annals of Statistic
Posterior Consistency via Precision Operators for Bayesian Nonparametric Drift Estimation in SDEs
We study a Bayesian approach to nonparametric estimation of the periodic
drift function of a one-dimensional diffusion from continuous-time data.
Rewriting the likelihood in terms of local time of the process, and specifying
a Gaussian prior with precision operator of differential form, we show that the
posterior is also Gaussian with precision operator also of differential form.
The resulting expressions are explicit and lead to algorithms which are readily
implementable. Using new functional limit theorems for the local time of
diffusions on the circle, we bound the rate at which the posterior contracts
around the true drift function
Semiparametric posterior limits
We review the Bayesian theory of semiparametric inference following Bickel
and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency
in parametric and semiparametric estimation problems, we consider the
Bernstein-von Mises theorem (see, e.g., Le Cam and Yang (1990)) and generalize
it to (LAN) regular and (LAE) irregular semiparametric estimation problems. We
formulate a version of the semiparametric Bernstein-von Mises theorem that does
not depend on least-favourable submodels, thus bypassing the most restrictive
condition in the presentation of Bickel and Kleijn (2012). The results are
applied to the (regular) estimation of the linear coefficient in partial linear
regression (with a Gaussian nuisance prior) and of the kernel bandwidth in a
model of normal location mixtures (with a Dirichlet nuisance prior), as well as
the (irregular) estimation of the boundary of the support of a monotone family
of densities (with a Gaussian nuisance prior).Comment: 47 pp., 1 figure, submitted for publication. arXiv admin note:
substantial text overlap with arXiv:1007.017
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