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An operational characterization of the notion of probability by algorithmic randomness and its applications | | Kohtaro Tadaki
; | | Date: |
18 Nov 2016 | | Abstract: | 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 of probability is not established yet. In this paper, based on
the toolkit of algorithmic randomness we present an operational
characterization of the notion of probability, called an ensemble. Algorithmic
randomness, also known as algorithmic information theory, is a field of
mathematics which enables us to consider the randomness of an individual
infinite sequence. We use the notion of Martin-Loef randomness with respect to
Bernoulli measure to present the operational characterization. As the first
step of the research of this line, in this paper we consider the case of finite
probability space, i.e., the case where the sample space of the underlying
probability space is finite, for simplicity. We give a natural operational
characterization of the notion of conditional probability in terms of ensemble,
and give equivalent characterizations of the notion of independence between two
events based on it. Furthermore, we give equivalent characterizations of the
notion of independence of an arbitrary number of events/random variables in
terms of ensembles. In particular, we show that the independence between
events/random variables is equivalent to the independence in the sense of van
Lambalgen’s Theorem, in the case where the underlying finite probability space
is computable. In the paper we make applications of our framework to
information theory and cryptography in order to demonstrate the wide
applicability of our framework to the general areas of science and technology. | | Source: | arXiv, 1611.6201 | | Services: | Forum | Review | PDF | Favorites | |
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