Related papers: Algorithmic information theory and martingales
The paper traces the development of the use of martingale methods in survival analysis from the mid 1970's to the early 1990's. This development was initiated by Aalen's Berkeley PhD-thesis in 1975, progressed through the work on estimation…
By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for…
A didactical survey of the foundations of Algorithmic Information Theory. These notes are short on motivation, history and background but introduce some of the main techniques and concepts of the field. The "manuscript" has been evolving…
The issue of defining a random sequence of qubits is studied in the framework of Algorithmic Free Probability Theory.Its connection with Quantum Algorithmic Information Theory is shown
Presents a history of the evolution of the author's ideas on program-size complexity and its applications to metamathematics over the course of more than four decades. Includes suggestions for further work.
In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on…
Considerable thought has been devoted to an adequate definition of the class of infinite, random binary sequences (the sort of sequence that almost certainly arises from flipping a fair coin indefinitely). The first mathematical exploration…
In this paper we study subsequences of random numbers. In Kamae (1973), selection functions that depend only on coordinates are studied, and their necessary and sufficient condition for the selected sequences to be normal numbers is given.…
This thesis details a class of partial orders on the space of probability distributions and the space of density operators which capture the idea of information content. Some links to domain theory and computational linguistics are also…
Information theory is concerned with the study of transmission, processing, extraction, and utilization of information. In its most abstract form, information is conceived as a means of resolving uncertainty. Shannon and Weaver (1949) were…
We survey concepts at the frontier of research connecting artificial, animal and human cognition to computation and information processing---from the Turing test to Searle's Chinese Room argument, from Integrated Information Theory to…
While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing…
This paper is a top down historical perspective on the several phases in the development of probability from its prehistoric origins to its modern day evolution, as one of the key methodologies in artificial intelligence, data science, and…
Free probability theory started in the 1980s has attracted much attention lately in signal processing and communications areas due to its applications in large size random matrices. However, it involves with massive mathematical concepts…
The notion of random sequence was introduced by Martin-Loef in 1966. At the same time he defined the so-called randomness deficiency function that shows how close are random sequences to non-random (in some natural sense). Other deficiency…
Given two events $A$ and $B$, Bayes' law is based on the argument that the probability of $A$ given $B$ is proportional to the probability of $B$ given $A$. When probabilities are interpreted in the Bayesian sense, Bayes' law constitutes a…
The classic model of computable randomness considers martingales that take real or rational values. Recent work by Bienvenu et al. (2012) and Teutsch (2014) shows that fundamental features of the classic model change when the martingales…
The history of computability theory and and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, \'Emil Borel discussed his ideas on computable real number functions in his introduction…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise,…
A concept of randomness for infinite time register machines (ITRMs), resembling Martin-L\"of-randomness, is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies…