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Cultural Differences between Materials Science and Image Processing
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Mary Comer, Charles A. Bouman, Jeffrey P. Simmons
Probability Axioms. The probability mapping has been generalized in terms of a set of axioms that allow much more freedom in the assignment of probabilities. These axioms are: P(A) ≥ 0 for any event A,P(S) = 1, where S is the set of all possible outcomes, andP(A ∪ B) = P(A) + P(B) for any events A and B for which A and B are disjoint sets.The property stated in Axiom (3) for two sets A and B also applies for a countably infinite number of sets that are pairwise disjoint.
Formal Reliability Analysis of Railway Systems Using Theorem Proving Technique
Published in Qamar Mahboob, Enrico Zio, Handbook of RAMS in Railway Systems, 2018
Waqar Ahmad, Osman Hasan, Sofiène Tahar
Mathematically, a measure space is defined as a triple (Ω, Σ, µ), where Ω is a set, called the sample space; Σ represents a σ algebra of subsets of Ω, where the subsets are usually referred to as measurable sets; and µ is a measure with domain Σ. A probability space is a measure space (Ω, Σ, Pr), such that the measure, referred to as the probability and denoted by Pr, of the sample space is 1. In the HOL formalization of probability theory [10], given a probability space p, the functions , , and return the corresponding Ω, Σ, and Pr, respectively. This formalization also includes the formal verification of some of the most widely used probability axioms, which play a pivotal role in formal reasoning about reliability properties.
Some Uncertain Decision Combinations
Published in Yunmin Zhu, Jie Zhou, Xiaojing Shen, Enbin Song, Yingting Luo, Networked Multisensor Decision and Estimation Fusion, 2012
Yunmin Zhu, Jie Zhou, Xiaojing Shen, Enbin Song, Yingting Luo
The most common representation of uncertainty used previously is obviously the probability, but it is not necessarily always the most appropriate one. For example, a coin with unknown bias has uncertain probabilities of head and tail. In many practical uncertain problems, the requirements of the probability defined in Kolmogorov’s probability axioms are not exactly satisfied. Therefore, an introduction of generalized measures is necessary to represent the uncertainty. So far, many other uncertainty representations—for example, Dempster–Shafer belief functions, possibility measures, ranking functions, and plausibility measures—have been considered in the literature. These measures do not satisfy the axiomatic definition of probability measure; instead they have their own axiomatic measure definitions, which are less restrictive than probability, i.e., relaxed probability measures. Among these, the plausibility measures are a very general representation of uncertainty (see Halpern (2003)). In fact, one can view probability measures, Dempster–Shafer belief functions, possibility measures, and ranking functions as special cases of plausibility measures.
Raising the Bar for Theories of Categorisation and Concept Learning: The Need to Resolve Five Basic Paradigmatic Tensions
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Ronaldo Vigo, Jay Wimsatt, Charles A. Doan, Derek E. Zeigler
To date, contrast cues and effects have occupied a consistent niche in categorisation research. Indeed, one could argue that the role of contrast has largely been default, ever-present without necessarily being a point of direct research concern. Consider the numerous varieties of probabilistic models across the field, ranging from exemplar models originating from or relating to the context model of Medin and Schaffer (1978) to Bayesian ‘updating’ models: the effectiveness of these models, by their inherent reliance on probability axioms, require the existence of alternative or contrasting categories. Embedded into the mathematics then is this notion of contrast, or a ‘one or the other’ categorisation mindset. One may then consider how the categorisation process may be impacted if clear contrast information is not provided to participants at all, and then how those models would account for data emerging from such an investigation.
A mosaic of Chu spaces and Channel Theory I: Category-theoretic concepts and tools
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2019
Chris Fields, James F. Glazebrook
Thus when the sequent’s conditional probability is , say, we have . A priori, one must have to apply in a argument. The probability of the former holding in , is . Then follows from the rule . Probability axioms for a Countable Classical Propositional Logic are developed in Allwein (2004) (cf. Allwein et al., 2004) to which we refer for details. Note that information flow in distributed systems can be interpreted dynamically; this amounts to causation in an informational context, consistent with the Dretskean nature of the theory. In this respect, the relations between information theory and logic are also conducive to understanding certain relations between causation and computation (Collier, 2011; Seligman, 2009).
Sustainability assessment of concrete bridge deck designs in coastal environments using neutrosophic criteria weights
Published in Structure and Infrastructure Engineering, 2020
Ignacio J. Navarro, Víctor Yepes, José V. Martí
(Liang, Wang, & Zhang, 2018; Ye, 2017) Let ā = 〈(a1,a2,a3); tā, iā, fā〉 and b̄ = 〈(b1,b2,b3); tb̄, ib̄, fb̄〉 be two single-valued triangular neutrosophic numbers. Let k be a real, positive number. Then, the basic arithmetic operations for neutrosophic numbers, based on Kolmogorov’s probability axioms, are defined as: