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The goal of this paper is to provide theorems on convergence rates of posterior distributions that can be applied to obtain good convergence rates in the context of density estimation as well as regression. We show how to choose priors so…

Most bandit algorithms assume that the reward variances or their upper bounds are known, and that they are the same for all arms. This naturally leads to suboptimal performance and higher regret due to variance overestimation. On the other…

Prior distributions for high-dimensional linear regression require specifying a joint distribution for the unobserved regression coefficients, which is inherently difficult. We instead propose a new class of shrinkage priors for linear…

We consider a situation where the distribution of a random variable is being estimated by the empirical distribution of noisy measurements of that variable. This is common practice in, for example, teacher value-added models and other…

We derive a conditional version of the classical regret-capacity theorem. This result can be used in universal prediction to find lower bounds on the minimal batch regret, which is a recently introduced generalization of the average regret,…

Bayesian predictive inference provides a coherent description of entire predictive uncertainty through predictive distributions. We examine several widely used sparsity priors from the predictive (as opposed to estimation) inference…

The prior distribution for the unknown model parameters plays a crucial role in the process of statistical inference based on Bayesian methods. However, specifying suitable priors is often difficult even when detailed prior knowledge is…

In this article, we study the binary classification problem with supervised data, in the case where the covariate-to-probability-of-success map is possibly spatially inhomogeneous. We devise nonparametric Bayesian procedures with…

As models of cognition grow in complexity and number of parameters, Bayesian inference with standard methods can become intractable, especially when the data-generating model is of unknown analytic form. Recent advances in simulation-based…

A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory…

In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non increasing densities in a Bayesian setting and derive concentration rate for the…

We revisit the classical credibility results of Jewell and B\"uhlmann to obtain credibility premiums for a GLM using a modern Bayesian approach. Here the prior distributions can be chosen without restrictions to be conjugate to the response…

Supremum norm loss is intuitively more meaningful to quantify function estimation error in statistics. In the context of multivariate nonparametric regression with unknown error, we propose a Bayesian procedure based on spike-and-slab prior…

A key sticking point of Bayesian analysis is the choice of prior distribution, and there is a vast literature on potential defaults including uniform priors, Jeffreys' priors, reference priors, maximum entropy priors, and weakly informative…

We develop a new approach to obtaining high probability regret bounds for online learning with bandit feedback against an adaptive adversary. While existing approaches all require carefully constructing optimistic and biased loss…

Modern approaches to perform Bayesian variable selection rely mostly on the use of shrinkage priors. That said, an ideal shrinkage prior should be adaptive to different signal levels, ensuring that small effects are ruled out, while keeping…

The practical implementation of Bayesian inference requires numerical approximation when closed-form expressions are not available. What types of accuracy (convergence) of the numerical approximations guarantee robustness and what types do…

Using data obtained in a controlled ad-auction experiment that we ran, we evaluate the regret-based approach to econometrics that was recently suggested by Nekipelov, Syrgkanis, and Tardos (EC 2015). We found that despite the weak…

In this paper we compare and contrast the behavior of the posterior predictive distribution to the risk of the maximum a posteriori estimator for the random features regression model in the overparameterized regime. We will focus on the…

Bayesian methods are actively used for parameter identification and uncertainty quantification when solving nonlinear inverse problems with random noise. However, there are only few theoretical results justifying the Bayesian approach.…