Related papers: Lessons from the German Tank Problem
Algorithmic statistics studies explanations of observed data that are good in the algorithmic sense: an explanation should be simple i.e. should have small Kolmogorov complexity and capture all the algorithmically discoverable regularities…
We consider the problem of identifying any $k$ out of the best $m$ arms in an $n$-armed stochastic multi-armed bandit. Framed in the PAC setting, this particular problem generalises both the problem of `best subset selection' and that of…
It is common for researchers to record long, multiple time series from experiments or calculations. But sometimes there are no good models for the systems or no applicable mathematical theorems that can tell us when there are basic…
Multi-armed bandit algorithms have been argued for decades as useful for adaptively randomized experiments. In such experiments, an algorithm varies which arms (e.g. alternative interventions to help students learn) are assigned to…
Two-sample testing is a fundamental problem in statistics. Despite its long history, there has been renewed interest in this problem with the advent of high-dimensional and complex data. Specifically, in the machine learning literature,…
Optimal prediction (OP) methods compensate for a lack of resolution in the numerical solution of complex problems through the use of an invariant measure as a prior measure in the Bayesian sense. In first-order OP, unresolved information is…
The problem of detecting changes in the statistical properties of a stochastic system and time series arises in various branches of science and engineering. It has a wide spectrum of important applications ranging from machine monitoring to…
Distributed estimation that recruits potentially large groups of humans to collect data about a phenomenon of interest has emerged as a paradigm applicable to a broad range of detection and estimation tasks. However, it also presents a…
We propose an analytical solution to the on-off problem within the framework of Bayesian statistics. Both the statistical significance for the discovery of new phenomena and credible intervals on model parameters are presented in a…
In statistical mechanics, the generally called Stirling approximation is actually an approximation of Stirling's formula. In this article, it is shown that the term that is dropped is in fact the one that takes fluctuations into account.…
The problem of statistical learning is to construct an accurate predictor of a random variable as a function of a correlated random variable on the basis of an i.i.d. training sample from their joint distribution. Allowable predictors are…
Inverse problems, where in broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific…
Despite recent successes in language models, their ability to represent numbers is insufficient. Humans conceptualize numbers based on their magnitudes, effectively projecting them on a number line; whereas subword tokenization fails to…
A puzzle about prisoners trying to identify the color of a hat on their head leads to a version where there are k more hats than prisoners. This generalized puzzle is related to the independence number of the arrangement graph A(m, n) and…
Given two random variables $X$ and $Y$, an operational approach is undertaken to quantify the ``leakage'' of information from $X$ to $Y$. The resulting measure $\mathcal{L}(X \!\! \to \!\! Y)$ is called \emph{maximal leakage}, and is…
In the multiarmed bandit problem a gambler chooses an arm of a slot machine to pull considering a tradeoff between exploration and exploitation. We study the stochastic bandit problem where each arm has a reward distribution supported in a…
Empirical Bayes methods use the data from parallel experiments, for instance, observations $X_k\sim\mathcal{N}(\Theta_k,1)$ for $k=1,2,\ldots,N$, to estimate the conditional distributions $\Theta_k|X_k$. There are two main estimation…
Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the…
Pattern recognition is a central topic in Learning Theory with numerous applications such as voice and text recognition, image analysis, computer diagnosis. The statistical set-up in classification is the following: we are given an i.i.d.…
We consider a situation where an agent has $T$ ressources to be allocated to a larger number $N$ of actions. Each action can be completed at most once and results in a stochastic reward with unknown mean. The goal of the agent is to…