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Vincze, I.; Hoffmann, K.; Wittwer, G.; Linares, Gabriela Castro; Warmuth, W.; Engelbert, H.J.; Lorenz, P.; Jäger,; Läuter, E.; Möhner, M.; Kirstein, B.M.

doi:10.1080/02331888108801588

Cited by 2 publications

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“…For the analysis of probabilistic systems, it is possible to delineate a principle which stands out from all others: the maximum probability ("MaxProb") principle [31][32][33][34][35]. This can be stated as:…”

Section: The Combinatorial Basismentioning

confidence: 99%

“…then by taking φ (·) = ln(·), κ = N −1 and the asymptotic limits N → ∞ and n i → ∞, ∀i (the "Stirling approximation"), D and H converge respectively to the Kullback-Leibler and Shannon functions (1)-(2) [33,34]. This provides a (well-known) justification for these functions, and their corresponding MinXEnt and MaxEnt principles, as a special case, independently of the arguments used in information theory.…”

Section: The Combinatorial Basismentioning

confidence: 99%

“…This study examines one such framework: the combinatorial (or probabilistic) basis of entropy, first given 130 years ago by Boltzmann [31] and subsequently promoted by Planck [32]. This involves the maximization of a governing probability distribution P or weight W of a system; this can be viewed as a generalized principle of probabilistic inference, aptly described by Vincze and Grendar & Grendar as the maximum probability ("MaxProb") principle [33,34]. It also leads to generalized definitions of cross-entropy and entropy, based purely on probability theory [35].…”

Section: Introductionmentioning

confidence: 99%

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Origins of the Combinatorial Basis of Entropy

Niven

2007

AIP Conference Proceedings

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The combinatorial basis of entropy, given by Boltzmann, can be written $H = N^{-1} \ln \mathbb{W}$, where $H$ is the dimensionless entropy, $N$ is the number of entities and $\mathbb{W}$ is number of ways in which a given realization of a system can occur (its statistical weight). This can be broadened to give generalized combinatorial (or probabilistic) definitions of entropy and cross-entropy: $H=\kappa (\phi(\mathbb{W}) +C)$ and $D=-\kappa (\phi(\mathbb{P}) +C)$, where $\mathbb{P}$ is the probability of a given realization, $\phi$ is a convenient transformation function, $\kappa$ is a scaling parameter and $C$ an arbitrary constant. If $\mathbb{W}$ or $\mathbb{P}$ satisfy the multinomial weight or distribution, then using $\phi(\cdot)=\ln(\cdot)$ and $\kappa=N^{-1}$, $H$ and $D$ asymptotically converge to the Shannon and Kullback-Leibler functions. In general, however, $\mathbb{W}$ or $\mathbb{P}$ need not be multinomial, nor may they approach an asymptotic limit. In such cases, the entropy or cross-entropy function can be {\it defined} so that its extremization ("MaxEnt'' or "MinXEnt"), subject to the constraints, gives the ``most probable'' (``MaxProb'') realization of the system. This gives a probabilistic basis for MaxEnt and MinXEnt, independent of any information-theoretic justification. This work examines the origins of the governing distribution $\mathbb{P}$.... (truncated)Comment: MaxEnt07 manuscript, version 4 revise

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Section: The Combinatorial Basismentioning

confidence: 99%

Section: The Combinatorial Basismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Origins of the Combinatorial Basis of Entropy

Niven

2007

AIP Conference Proceedings

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“…Over the past century, maximum entropy (MaxEnt) methods have been developed for the construction of probabilistic models, initially in thermodynamics [5,6] and subsequently for all probabilistic systems [7,8,9]. Although imbued with several information-theoretic interpretations [7,8,9,10], the success of such models rests ultimately on the maximum probability (MaxProb) principle of Boltzmann [5,6,11,12,13,14,15,16,17,18]: "a system can be represented by its most probable state." This provides a probabilistic definition of the (relative) entropy function:…”

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confidence: 99%

Maximum-entropy weighting of multiple earth climate models

Niven

2011

Clim Dyn

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A maximum entropy-based framework is presented for the synthesis of projections from multiple Earth climate models. This identifies the most representative (most probable) model from a set of climate models -as defined by specified constraints -eliminating the need to calculate the entire set. Two approaches are developed, based on individual climate models or ensembles of models, subject to a single cost (energy) constraint or competing cost-benefit constraints. A finitetime limit on the minimum cost of modifying a model synthesis framework, at finite rates of change, is also reported.Keywords climate model · maximum entropy method · Boltzmann principle · thermodynamics · cost-benefit analysis · finite-time information limit 1 Introduction A major challenge facing humanity is the possibility of climate change due to human and/or natural forcings, and how best to respond in a rational and informed manner. To this end, detailed global circulation models (GCMs) have been developed to predict the behaviour of the Earth climate system (atmosphere and oceans), involving solution of the continuity, Navier-Stokes, angular momentum and energy equations and constitutive

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Book reviews

Cited by 2 publications

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Origins of the Combinatorial Basis of Entropy

Origins of the Combinatorial Basis of Entropy

Maximum-entropy weighting of multiple earth climate models

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