Related papers: Algorithmic information theory and martingales

Beginning in the 1970s, statistician-cum-logician Per Martin-L\"of wrote a series of papers developing what became Martin-L\"of type theory, realizing a system where the distinction between mathematics and programming disappears. Inspired…

TThe problem is to identify a probability associated with a set of natural numbers, given an infinite data sequence of elements from the set. If the given sequence is drawn i.i.d. and the probability mass function involved (the target)…

Algorithmic theories of randomness can be related to theories of probabilistic sequence prediction through the notion of a predictor, defined as a function which supplies lower bounds on initial-segment probabilities of infinite sequences.…

In the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the…

These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. The first part of the notes gives an introduction to probability theory. It explains the notion of random events and random variables,…

This is a 1971 dissertation. Only its extended abstract was published at the time. While some results appeared in other publications, a number of details remained unpublished and may still have relevance.

The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search…

Randomness in the sense of Martin-L\"of can be defined in terms of lower semicomputable supermartingales. We show that such a supermartingale cannot be replaced by a pair of supermartingales that bet only on the even bits (the first one)…

Shannon based his information theory on the notion of probability measures as it we developed by Kolmogorov. In this paper we study some fundamental problems in information theory based on expectation measures. In the theory of expectation…

Pseudorandmness plays an important role in number theory, complexity theory and cryptography. Our aim is to use models of arithmetic to explain pseudorandomness by randomness. To this end we construct a set of models $\cal M$, a common…

A selection of the relevant theorems of Probability Theory that comes directly from Kolmogorov's axioms, Set Theory basic results, definitions and rules of inference are listed and proven in a systematic approach, aiming the student who…

Inductive inference is a recursion-theoretic theory of learning, first developed by E. M. Gold (1967). This paper surveys developments in probabilistic inductive inference. We mainly focus on finite inference of recursive functions, since…

Given the constant rise in quantity and quality of data obtained from neural systems on many scales ranging from molecular to systems', information-theoretic analyses became increasingly necessary during the past few decades in the…

This chapter discusses the institutional approach for organizing and maintaining ontologies. The theory of institutions was named and initially developed by Joseph Goguen and Rod Burstall. This theory, a metatheory based on category theory,…

We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a…

Kolmogorov argued that the concept of information exists also in problems with no underlying stochastic model (as Shannon's information representation) for instance, the information contained in an algorithm or in the genome. He introduced…

Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and…

The random forest algorithm, proposed by L. Breiman in 2001, has been extremely successful as a general-purpose classification and regression method. The approach, which combines several randomized decision trees and aggregates their…

The designation ``Bernstein-von Mises theorem'' is apparently due to Lucien Le Cam. Roughly, the assertion of this theorem states that the posterior distribution of a parameter, conditioned on a large sample, is approximately normal,…

John organized a state lottery and his wife won the main prize. You may feel that the event of her winning wasn't particularly random, but how would you argue that in a fair court of law? Traditional probability theory does not even have…