Large Deviations and Idempotent Probability

This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as a result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values.This paper is a brief introduction, practically without exact theorems and proofs, to the Maslov dequantization and idempotent and tropical mathematics. Our list of references is not complete (not at all). Additional references can be found, e.g., in the electronic archive http://arXiv.org

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“…For relations and applications to large deviations see [1,[35][36][37][38]142] and especially the book of A. A. Puhalskii [141].…”

Section: Convolution and The Fourier-legendre Transformmentioning

confidence: 99%

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

Litvinov

2007

J Math Sci

140

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“…There are finally close connections between possibility theory and idempotent analysis [113]. The theory of large deviations in probability theory [123] also handles set-functions that look like possibility measures [117]. Here we focus on the role of possibility theory in the theory of imprecise probability.…”

Section: Quantitative Possibility Theorymentioning

confidence: 99%

Possibility Theory and Its Applications: Where Do We Stand?

Dubois

Prade

2015

Springer Handbook of Computational Intelligence

148

101

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This paper provides an overview of possibility theory, emphasizing its historical roots and its recent developments. Possibility theory lies at the crossroads between fuzzy sets, probability and non-monotonic reasoning. Possibility theory can be cast either in an ordinal or in a numerical setting. Qualitative possibility theory is closely related to belief revision theory, and common-sense reasoning with exception-tainted knowledge in Artificial Intelligence. Possibilistic logic provides a rich representation setting, which enables the handling of lower bounds of possibility theory measures, while remaining close to classical logic. Qualitative possibility theory has been axiomatically justified in a decision-theoretic framework in the style of Savage, thus providing a foundation for qualitative decision theory. Quantitative possibility theory is the simplest framework for statistical reasoning with imprecise probabilities. As such it has close connections with random set theory and confidence intervals, and can provide a tool for uncertainty propagation with limited statistical or subjective information.

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“…One may define a probability measure with respect to the max-plus algebra [1,7,19,21]. Let the pair (Ω, F) denote a sample space and associated sigma-algebra of sets on that space.…”

Section: A Review Of Max-plus Probabilitymentioning

confidence: 99%