WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 15, 2016
On Probability Algebra: Classic Theory of Probability Revisited
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Abstract: It is recognized that the classic probability theory is cyclically defined among a small set of coupled operations. The sample space of probability is not merely 1-D invariant structures rather than n-D variant hyperstructure where the types of probability events encompassing those of joint or disjoint as well as dependent, independent, or mutually-exclusive ones. These fundamental properties of probability lead to a three-dimensional dynamic model of probability structures constrained by types of sample spaces, relations and dependencies of events. A reductive framework of general probability theory is rigorously derived from the independently defined model of conditional probability. This basic study reveals that the Bayes’ law needs to be extended in order to fit more general contexts of variant sample spaces and complex event properties. The revisited probability theory enables an extended mathematical structure known as probability algebra for rigorous manipulating uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft computing, semantic computing, and brain informatics.
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Keywords: Denotational mathematics, probability theory, probability algebra, statistics, formal inference, cognitive informatics, cognitive computing, computational intelligence, semantic computing, soft computing, brain informatics, cognitive systems
Pages: 550-565
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #52