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This work studies the distributed learning process on a network of agents. Agents make partial observation about an unknown hypothesis and iteratively share their beliefs over a set of possible hypotheses with their neighbors to learn the…
A popular scientific contribution should not contradict any established facts and ought to be understandable. I complied with both these requirements and am offering a sufficiently full introduction to probability theory. Furthermore, I…
This paper has been withdrawn, not because of any errors (that we know of), but because rather than presenting our material as a series of 3 papers, as we originally intended, we have now combined them into one long paper, which is "A…
A reply to the commentaries of Yana (2013), and some jots on information theory.
We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts…
The Annals of Applied Probability 16 (2006) 984--1033 [URL: http://projecteuclid.org/euclid.aoap/1151592257]
These lecture notes have been developed for the course Computational Social Choice of the Artificial Intelligence MSc programme at the University of Groningen. They cover mathematical and algorithmic aspects of voting theory.
Information theory is a statistical theory dealing with the relative state of detectors and physical systems. Because of this physicality of information, the classical framework of Shannon needs to be extended to deal with quantum…
Recently people started to understand that applications of the mathematical formalism of quantum theory are not reduced to physics. Nowadays, this formalism is widely used outside of quantum physics, in particular, in cognition, psychology,…
The framework of algorithmic knowledge assumes that agents use deterministic knowledge algorithms to compute the facts they explicitly know. We extend the framework to allow for randomized knowledge algorithms. We then characterize the…
This is an expository article on the Poisson binomial distribution. We review lesser known results and recent progress on this topic, including geometry of polynomials and distribution learning. We also provide examples to illustrate the…
The classical propositional assumption-based model is extended to incorporate probabilities for the assumptions. Then it is placed into the framework of evidence theory. Several authors like Laskey, Lehner (1989) and Provan (1990) already…
This is a 20-year old review on singularities and singularity theorems. The main reason to submit it now is -apart from increasing its availability- to correct a very strange error that appears in the journal's online version: it contains…
The paper has been withdrawn because the research work is still in progress.
This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern…
The main purpose of this paper is to correct an error in the previously submitted version [*] := arXiv:2004.13749v1. [*] had been already accepted for publication in a scientific journal, but withdrawn by the author after the discovery of…
In this article we review Tononi's (2008) theory of consciousness as integrated information. We argue that previous formalizations of integrated information (e.g. Griffith, 2014) depend on information loss. Since lossy integration would…
The progress of machine learning over the past decade is undeniable. In retrospect, it is both remarkable and unsettling that this progress was achievable with little to no rigorous theory to guide experimentation. Despite this fact,…
This paper gives game-theoretic versions of several results on "merging of opinions" obtained in measure-theoretic probability and algorithmic randomness theory. An advantage of the game-theoretic versions over the measure-theoretic results…
In the footsteps of the book \textit{Measure Theory and Integration By and For the Learner} of our series in Probability Theory and Statistics, we intended to devote a special volume of the very probabilistic aspects of the first cited…