50,303 research outputs found
Categorical Bayesian Inference
The language and constructions of category theory have proven useful in unifying disparate fields of study and bridging formal gaps between approaches, so it is natural that a categorial eye should be turned to the theory of probability and its relation to formal logic. Continuing from the foundational work of Lawvere and Giry in developing a functorial theory of probability, Stuartz and Culbertson detail the central importance of and connection between deterministic processes and stochastic processes. Fong expanded this theory to give a categorical account of Bayesian causality. Here we collect and summarize the rich body of research in categorical probability theory, and further develop mathematical machinery for applications in algorithmic Bayesian statistics
Unifying Logic and Probability: A New Dawn for AI?
Abstract. Logic and probability theory are two of the most important branches of mathematics and each has played a significant role in artificial intelligence (AI) research. Beginning with Leibniz, scholars have attempted to unify logic and probability. For "classical" AI, based largely on first-order logic, the purpose of such a unification is to handle uncertainty and facilitate learning from real data; for "modern" AI, based largely on probability theory, the purpose is to acquire formal languages with sufficient expressive power to handle complex domains and incorporate prior knowledge. This paper provides a brief summary of an invited talk describing efforts in these directions, focusing in particular on open-universe probability models that allow for uncertainty about the existence and identity of objects
A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS
In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, which rigorously defines the infinitesimals
Incompatible Multiple Consistent Sets of Histories and Measures of Quantumness
In the consistent histories (CH) approach to quantum theory probabilities are
assigned to histories subject to a consistency condition of negligible
interference. The approach has the feature that a given physical situation
admits multiple sets of consistent histories that cannot in general be united
into a single consistent set, leading to a number of counter-intuitive or
contrary properties if propositions from different consistent sets are combined
indiscriminately. An alternative viewpoint is proposed in which multiple
consistent sets are classified according to whether or not there exists any
unifying probability for combinations of incompatible sets which replicates the
consistent histories result when restricted to a single consistent set. A
number of examples are exhibited in which this classification can be made, in
some cases with the assistance of the Bell, CHSH or Leggett-Garg inequalities
together with Fine's theorem. When a unifying probability exists logical
deductions in different consistent sets can in fact be combined, an extension
of the "single framework rule". It is argued that this classification coincides
with intuitive notions of the boundary between classical and quantum regimes
and in particular, the absence of a unifying probability for certain
combinations of consistent sets is regarded as a measure of the "quantumness"
of the system. The proposed approach and results are closely related to recent
work on the classification of quasi-probabilities and this connection is
discussed.Comment: 29 pages. Second revised version with discussion of the sample space
and non-uniqueness of the unifying probability and small errors correcte
A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS - 6th ed.
It was a surprise for me when in 1995 I received a manuscript from the mathematician, experimental writer and innovative painter Florentin Smarandache, especially because the treated subject was of philosophy - revealing paradoxes - and logics. He had generalized the fuzzy logic, and introduced two new concepts: a) âneutrosophyâ â study of neutralities as an extension of dialectics; b) and its derivative âneutrosophicâ, such as âneutrosophic logicâ, âneutrosophic setâ, âneutrosophic probabilityâ, and âneutrosophic statisticsâ and thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics. It was known to me his setting up in 1980âs of a new literary and artistic avant-garde movement that he called âparadoxismâ, because I received some books and papers dealing with it in order to review them for the German journal âZentralblatt fur Mathematikâ. It was an inspired connection he made between literature/arts and science, philosophy. We started a long correspondence with questions and answers. Because paradoxism supposes multiple value sentences and procedures in creation, antisense and non-sense, paradoxes and contradictions, and itâs tight with neutrosophic logic, I would like to make a small presentation
A Survey of Quantum Theory Inspired Approaches to Information Retrieval
Since 2004, researchers have been using the mathematical framework of Quantum Theory (QT) in Information Retrieval (IR). QT offers a generalized probability and logic framework. Such a framework has been shown capable of unifying the representation, ranking and user cognitive aspects of IR, and helpful in developing more dynamic, adaptive and context-aware IR systems. Although Quantum-inspired IR is still a growing area, a wide array of work in different aspects of IR has been done and produced promising results. This paper presents a survey of the research done in this area, aiming to show the landscape of the field and draw a road-map of future directions
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