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Volume 13: Number 4: Article 2
Basic Elements and Problems of Probability Theory
Hans Primas, Laboratory of Physical Chemistry, ETH-Zentrum, CH-8092
Zürich, Switzerland
After a brief review of ontic and epistemic descriptions, and of subjective,
logical and statistical interpretations of probability, we summarize
the traditional axiomatization of calculus of probability in terms of
Boolean algebras and its set-theoretical realization in terms of Kolmogorov
probability spaces. Since the axioms of mathematical probability theory
say nothing about the conceptual meaning of "randomness" one considers
probability as property of the generating conditions of a process so
that one can relate randomness with predictability (or retrodictability).
In the measure-theoretical codification of stochastic processes genuine
chance processes can be defined rigorously as so-called regular processes
which do not allow a long-term prediction. We stress that stochastic
processes are equivalence classes of individual point functions so that
they do not refer to individual processes but only to an ensemble of
statistically equivalent individual processes. Less popular but conceptually
more important than statistical descriptions are individual descriptions
which refer to individual chaotic processes. First, we review the individual
description based on the generalized harmonic analysis by Norbert Wiener.
It allows the definition of individual purely chaotic processes which
can be interpreted as trajectories of regular statistical stochastic
processes. Another individual description refers to algorithmic procedures
which connect the intrinsic randomness of a finite sequence with the
complexity of the shortest program necessary to produce the sequence.
Finally, we ask why there can be laws of chance. We argue that random
events fulfill the laws of chance if and only if they can be reduced
to (possibly hidden) deterministic events. This mathematical result
may elucidate the fact that not all non-predictable events can be grasped
by the methods of mathematical probability theory.
Keywords: probability, stochasticity, chaos, randomness, chance, determinism
FULL TEXT:
Basic Elements and Problems of Probability
Theory
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