25 research outputs found
Adaptive Bernstein-von Mises theorems in Gaussian white noise
We investigate Bernstein-von Mises theorems for adaptive nonparametric
Bayesian procedures in the canonical Gaussian white noise model. We consider
both a Hilbert space and multiscale setting with applications in and
respectively. This provides a theoretical justification for plug-in
procedures, for example the use of certain credible sets for sufficiently
smooth linear functionals. We use this general approach to construct optimal
frequentist confidence sets based on the posterior distribution. We also
provide simulations to numerically illustrate our approach and obtain a visual
representation of the geometries involved.Comment: 48 pages, 5 figure
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
A Bayesian nonparametric approach to log-concave density estimation
The estimation of a log-concave density on is a canonical
problem in the area of shape-constrained nonparametric inference. We present a
Bayesian nonparametric approach to this problem based on an exponentiated
Dirichlet process mixture prior and show that the posterior distribution
converges to the log-concave truth at the (near-) minimax rate in Hellinger
distance. Our proof proceeds by establishing a general contraction result based
on the log-concave maximum likelihood estimator that prevents the need for
further metric entropy calculations. We also present two computationally more
feasible approximations and a more practical empirical Bayes approach, which
are illustrated numerically via simulations.Comment: 39 pages, 17 figures. Simulation studies were significantly expanded
and one more theorem has been adde
The Le Cam distance between density estimation, Poisson processes and Gaussian white noise
It is well-known that density estimation on the unit interval is
asymptotically equivalent to a Gaussian white noise experiment, provided the
densities have H\"older smoothness larger than and are uniformly bounded
away from zero. We derive matching lower and constructive upper bounds for the
Le Cam deficiencies between these experiments, with explicit dependence on both
the sample size and the size of the densities in the parameter space. As a
consequence, we derive sharp conditions on how small the densities can be for
asymptotic equivalence to hold. The related case of Poisson intensity
estimation is also treated.Comment: Some results from an earlier version of this preprint have been moved
to arXiv:1802.0342
Pointwise uncertainty quantification for sparse variational Gaussian process regression with a Brownian motion prior
We study pointwise estimation and uncertainty quantification for a sparse
variational Gaussian process method with eigenvector inducing variables. For a
rescaled Brownian motion prior, we derive theoretical guarantees and
limitations for the frequentist size and coverage of pointwise credible sets.
For sufficiently many inducing variables, we precisely characterize the
asymptotic frequentist coverage, deducing when credible sets from this
variational method are conservative and when overconfident/misleading. We
numerically illustrate the applicability of our results and discuss connections
with other common Gaussian process priors.Comment: 24 pages, 1 figure, to appear in Advances in Neural Information
Processing Systems 37 (NeurIPS 2023
Debiased Bayesian inference for average treatment effects
Bayesian approaches have become increasingly popular in causal inference
problems due to their conceptual simplicity, excellent performance and in-built
uncertainty quantification ('posterior credible sets'). We investigate Bayesian
inference for average treatment effects from observational data, which is a
challenging problem due to the missing counterfactuals and selection bias.
Working in the standard potential outcomes framework, we propose a data-driven
modification to an arbitrary (nonparametric) prior based on the propensity
score that corrects for the first-order posterior bias, thereby improving
performance. We illustrate our method for Gaussian process (GP) priors using
(semi-)synthetic data. Our experiments demonstrate significant improvement in
both estimation accuracy and uncertainty quantification compared to the
unmodified GP, rendering our approach highly competitive with the
state-of-the-art.Comment: NeurIPS 201
Bayesian estimation in a multidimensional diffusion model with high frequency data
We consider nonparametric Bayesian inference in a multidimensional diffusion
model with reflecting boundary conditions based on discrete high-frequency
observations. We prove a general posterior contraction rate theorem in
-loss, which is applied to Gaussian priors. The resulting posteriors, as
well as their posterior means, are shown to converge to the ground truth at the
minimax optimal rate over H\"older smoothness classes in any dimension. Of
independent interest and as part of our proofs, we show that certain
frequentist penalized least squares estimators are also minimax optimal.Comment: 56 pages, 1 figur
Debiased Bayesian inference for average treatment effects
No abstract availabl