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We introduce the Clifford entropy, a measure of how close an arbitrary unitary is to a Clifford unitary, which generalizes the stabilizer entropy for states. We show that this quantity vanishes if and only if a unitary is Clifford, is…

We show, how the classical Berry-Esseen theorem for normal approximation may be used to derive rates of convergence for random sums of centerd, real-valued random variables with respect to a certain class of probability metrics, including…

In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability…

In this paper, we refine the Berry-Esseen bounds for the multivariate normal approximation of Polyak-Ruppert averaged iterates arising from the linear stochastic approximation (LSA) algorithm with decreasing step size. We consider the…

In this paper, we introduce a method for approximating the solution to inference and optimization tasks in uncertain and deterministic reasoning. Such tasks are in general intractable for exact algorithms because of the large number of…

Probabilistic proofs of the Johnson-Lindenstrauss lemma imply that random projection can reduce the dimension of a data set and approximately preserve pairwise distances. If a distance being approximately preserved is called a success, and…

Which classes can be learned properly in the online model? -- that is, by an algorithm that at each round uses a predictor from the concept class. While there are simple and natural cases where improper learning is necessary, it is natural…

Charles L. Dodgson, also known as Lewis Carroll, in his book "Pillow problems" from 1893 asked for the likelihood of a random triangle to be obtuse. Clearly, the answer to Dodgson's question depends strongly on the assumed random…

We consider interval estimation of the difference between two binomial proportions. Several methods of constructing such an interval are known. Unfortunately those confidence intervals have poor coverage probability: it is significantly…

A well-known technique in estimating probabilities of rare events in general and in information theory in particular (used, e.g., in the sphere-packing bound), is that of finding a reference probability measure under which the event of…

This document presents the statistical methods used to process low-level measurements in the presence of noise. These methods can be classical or Bayesian. The question is placed in the general framework of the problem of nuisance…

Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson…

The goal of a learning algorithm is to receive a training data set as input and provide a hypothesis that can generalize to all possible data points from a domain set. The hypothesis is chosen from hypothesis classes with potentially…

We revisit the classical problem of universal prediction of stochastic sequences with a finite time horizon $T$ known to the learner. The question we investigate is whether it is possible to derive vanishing regret bounds that hold with…

The Chernoff bound is a well-known tool for obtaining a high probability bound on the expectation of a Bernoulli random variable in terms of its sample average. This bound is commonly used in statistical learning theory to upper bound the…

Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we…

INTRODUCTION: Wald's, the likelihood ratio (LR) and Rao's score tests and their corresponding confidence intervals (CIs), are the three most common estimators of parameters of Generalized Linear Models. On finite samples, these estimators…

We derive simple but nearly tight upper and lower bounds for the binomial lower tail probability (with straightforward generalization to the upper tail probability) that apply to the whole parameter regime. These bounds are easy to compute…

In modern scientific research, small-scale studies with limited participants are increasingly common. However, interpreting individual outcomes can be challenging, making it standard practice to combine data across studies using random…

We consider the problem of identifying the parameters of an unknown mixture of two arbitrary $d$-dimensional gaussians from a sequence of independent random samples. Our main results are upper and lower bounds giving a computationally…