35 research outputs found
Kolyan Ray and Botond Szabo’s contribution to the Discussion of “Martingale Posterior Distributions” by Fong, Holmes and Walke
No full textDiscussion of the article: martingale posterior distributions by Fong et al
Variational Bayes for High-Dimensional Linear Regression With Sparse Priors
No full textWe study a mean-field spike and slab variational Bayes (VB) approximation to Bayesian model selection priors in sparse high-dimensional linear regression. Under compatibility conditions on the design matrix, oracle inequalities are derived for the mean-field VB approximation, implying that it converges to the sparse truth at the optimal rate and gives optimal prediction of the response vector. The empirical performance of our algorithm is studied, showing that it works comparably well as other state-of-the-art Bayesian variable selection methods. We also numerically demonstrate that the widely used coordinate-ascent variational inference algorithm can be highly sensitive to the parameter updating order, leading to potentially poor performance. To mitigate this, we propose a novel prioritized updating scheme that uses a data-driven updating order and performs better in simulations. The variational algorithm is implemented in the R package sparsevb. Supplementary materials for this article are available online.</p
Spike and slab variational Bayes for high dimensional logistic regression
No full textNo abstract availabl
A Bayesian nonparametric approach to log-concave density estimation
No full textThe estimation of a log-concave density on R 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 further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations
Contributed Discussion to "Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects"
No full textDiscussion of Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding and Heterogeneous Effects by Hahn, Murray & Carvalho. Our discussion can be found on page 1026-102
Adaptive Bernstein–von Mises theorems in Gaussian white noise
No full textWe 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 L2
and L∞, 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
Nonparametric Bayesian estimation in a multidimensional diffusion model with high frequency data
No full textWe 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: 61 pages, 1 figur
Pointwise uncertainty quantification for sparse variational Gaussian process regression with a Brownian motion prior
No full textWe 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
The Le Cam distance between density estimation, Poisson processes and Gaussian white noise
No full textIt is well known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities have Hölder smoothness larger than 1/2 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
