相关论文: An operational characterization of the notion of p…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
We study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences. We explore the following problems: monotone complexity on product space, Lambalgen's theorem for correlated probability,…
We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a…
The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These…
We consider the first-order theory of random variables with the probabilistic independence relation, which concerns statements consisting of random variables, the probabilistic independence symbol, logical operators, and existential and…
This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We…
The paper considers quantitative versions of different randomness notions: algorithmic test measures the amount of non-randomness (and is infinite for non-random sequences). We start with computable measures on Cantor space (and Martin-Lof…
An a priori semimeasure (also known as "algorithmic probability" or "the Solomonoff prior" in the context of inductive inference) is defined as the transformation, by a given universal monotone Turing machine, of the uniform measure on the…
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure…
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…
Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them…
The definition of conditional probability in case of continuous distributions was an important step in the development of mathematical theory of probabilities. How can we define this notion in algorithmic probability theory? In this survey…
This paper is a comment on the paper "Quantum Mechanics and Algorithmic Randomness" was written by Ulvi Yurtsever \cite{Yurtsever} and the briefly explanation of the algorithmic randomness of quantum measurements results. There are…
Though the ability of human beings to deal with probabilities has been put into question, the assessment of rarity is a crucial competence underlying much of human decision-making and is pervasive in spontaneous narrative behaviour. This…
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure…
This work starts from definition of randomness, the results of algorithmic randomness are analyzed from the perspective of application. Then, the source and nature of randomness is explored, and the relationship between infinity and…
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.…
A random set is a generalisation of a random variable, i.e. a set-valued random variable. The random set theory allows a unification of other uncertainty descriptions such as interval variable, mass belief function in Dempster-Shafer theory…
The concept of typicality refers to properties holding for the "overwhelming majority" of cases and is a fundamental idea of the qualitative approach to dynamical problems. We argue that measure-theoretical typicality would be the adequate…
This text presents an unified approach of probability and statistics in the pursuit of understanding and computation of randomness in engineering or physical or social system with prediction with generalizability. Starting from elementary…