Related papers: Pascal, Fermat et la g\'eom\'etrie du hasard
These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the dynamics of rational maps on P^1. We first…
Decision theories offer principled methods for making choices under various types of uncertainty. Algorithms that implement these theories have been successfully applied to a wide range of real-world problems, including materials and drug…
The primary sourcebook for developments based on the data of the world components "Theory of Intellectualities and Mathematical Statistics" (TIMS) collections of the Department of Mathematics, Physics and Astronomy of Odessky National…
The integration of the history and philosophy of statistics was initiated at least by Hacking (1975) and advanced by Hacking (1990), Mayo (1996), and Zabell (2005), but it has not received sustained follow-up. Yet such integration is more…
What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much much…
Bayesian statistics is based on the subjective definition of probability as {\it ``degree of belief''} and on Bayes' theorem, the basic tool for assigning probabilities to hypotheses combining {\it a priori} judgements and experimental…
We re-evaluate the great Leibniz-Newton calculus debate, exactly three hundred years after it culminated, in 1712. We reflect upon the concept of invention, and to what extent there were indeed two independent inventors of this new…
The ideas of aleatoric and epistemic uncertainty are widely used to reason about the probabilistic predictions of machine-learning models. We identify incoherence in existing discussions of these ideas and suggest this stems from the…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
This PhD thesis deals with a number of different problems in mathematical physics with the common thread that they have probabilistic aspects. The problems all stem from mathematical studies of lattice systems in statistical and quantum…
The curse of dimensionality is a common phenomenon which affects analysis of datasets characterized by large numbers of variables associated with each point. Problematic scenarios of this type frequently arise in classification algorithms…
The three key documents for study geometry are: 1) "The Elements" of Euclid, 2) the lecture by B. Riemann at G\"ottingen in 1854 entitled "\"Uber die Hypothesen welche der Geometrie zu Grunde liegen" (On the hypotheses which underlie…
A family of random probabilities is defined and studied. This family contains the Dirichlet process as a special case, corresponding to an inner point in the appropriate parameter space. The extension makes it possible to have random means…
This is a joint introduction to classical and free probability, which are twin sisters. We first review the foundations of classical probability, notably with the main limiting theorems (CLT, CCLT, PLT, CPLT), and with a look into examples…
We argue using simple models that all successful practical uses of probabilities originate in quantum fluctuations in the microscopic physical world around us, often propagated to macroscopic scales. Thus we claim there is no physically…
When collaborating with an AI system, we need to assess when to trust its recommendations. If we mistakenly trust it in regions where it is likely to err, catastrophic failures may occur, hence the need for Bayesian approaches for…
Invitation to the statistical study of language: The topic of this presentation is the interdisciplinary nexus between linguistics and statistics. It targets linguists, for whom it may have a theoretical interest, or professionals that work…
I discuss the meaning of probability in the Everett-Wheeler interpretation of quantum mechanics, together with the problem of defining histories. To resolve these, I propose an understanding of probability arising from a form of temporal…
Triggered by a recent interesting New Scientist article on the too frequent incorrect use of probabilistic evidence in courts, I introduce the basic concepts of probabilistic inference with a toy model, and discuss several important issues…
An informal and elementary introduction to probability scoring and forecast verification and improvement, slightly extended from Significance 22:3(2025)16, which might be useful for less mathematical readers as a prologue to the classic…