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Axioms 2012, 1(1), 38-73; doi:10.3390/axioms1010038

Foundations of Inference

1,*  and 2
1 Departments of Physics and Informatics, University at Albany (SUNY), Albany, NY 12222, USA 2 Maximum Entropy Data Consultants Ltd., Kenmare, County Kerry, Ireland
* Author to whom correspondence should be addressed.
Received: 20 January 2012 / Revised: 1 June 2012 / Accepted: 7 June 2012 / Published: 15 June 2012
(This article belongs to the Special Issue Axioms: Feature Papers)
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We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general lattice symmetries. With other applications such as measure theory in mind, our derivations assume minimal symmetries, relying on neither negation nor continuity nor differentiability. Each relevant symmetry corresponds to an axiom of quantification, and these axioms are used to derive a unique set of quantifying rules that form the familiar probability calculus. We also derive a unique quantification of divergence, entropy and information.
Keywords: measure; divergence; probability; information; entropy; lattice measure; divergence; probability; information; entropy; lattice
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Knuth, K.H.; Skilling, J. Foundations of Inference. Axioms 2012, 1, 38-73.

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