Lattices and Their Consistent Quantification

Knuth, Kevin H.

doi:10.1002/andp.201700370

Cited by 8 publications

(24 citation statements)

References 51 publications

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“…Partitioning and combination can be filled out in a Boolean lattice defined by join ∨ (logical OR) which upgrades ⊕ to nondisjoint components, and meet ∧ (logical AND) which identifies overlaps. [14,15] In this context, we recognize p(B A) as the transition probability from source A to destination B and traditionally written as Pr(B | A). Symmetry of the product rule then gives Bayes' theorem…”

Section: Bayes' Theoremmentioning

confidence: 99%

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The Symmetrical Foundation of Measure, Probability, and Quantum Theories

Skilling¹,

Knuth

2018

Annalen der Physik

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Quantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are ubiquitous. Specifically, measure theory formalizes addition, and probability theory formalizes inference in terms of proportions. Quantum theory rests on the same simple symmetries, but is formalized in two dimensions, not just one, in order to track an object through its binary interactions with other objects. The symmetries still require sum and product rules (here known as the Feynman rules), but they apply to complex numbers instead of real scalars, with observable probabilities being modulus squared (known as the Born rule). The standard quantum formalism follows. There is no mystery or weirdness, just ordinary probabilistic inference.

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Section: Bayes' Theoremmentioning

confidence: 99%

“…Paths obey the symmetries listed above, to which their representations must conform. Accordingly, their associated scalars obey the standard product rule (15)…”

Section: The Product Rulesmentioning

confidence: 99%

The Symmetrical Foundation of Measure, Probability, and Quantum Theories

Skilling¹,

Knuth

2018

Annalen der Physik

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“…This can easily be checked for three and four variables by direct calculation, and by referring to the group Table 1. Equations (18) and (19) are seen to relate functions of the higher lattice elements to functions of the join irreducible elements.…”

Section: Symmetries Reveal a Wide Range Of New Relationsmentioning

confidence: 99%

“…By applying the down-set operator,m, to π we generate some interesting relations. As seen in Equations (18), the result of this operation is the delta, the conditional interaction information,…”

Section: Conditional Log Likelihoods and Deltasmentioning

confidence: 99%

Symmetries among Multivariate Information Measures Explored Using Möbius Operators

Galas

Sakhanenko

2019

Entropy

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Relations between common information measures include the duality relations based on Möbius inversion on lattices, which are the direct consequence of the symmetries of the lattices of the sets of variables (subsets ordered by inclusion). In this paper we use the lattice and functional symmetries to provide a unifying formalism that reveals some new relations and systematizes the symmetries of the information functions. To our knowledge, this is the first systematic examination of the full range of relationships of this class of functions. We define operators on functions on these lattices based on the Möbius inversions that map functions into one another, which we call Möbius operators, and show that they form a simple group isomorphic to the symmetric group S3. Relations among the set of functions on the lattice are transparently expressed in terms of the operator algebra, and, when applied to the information measures, can be used to derive a wide range of relationships among diverse information measures. The Möbius operator algebra is then naturally generalized which yields an even wider range of new relationships.

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“…The laws governing probabilities themselves seem to reflect that of more elementary mathematical structures. Kevin Knuth shows how these emerge from lattice theory, which deals with relations in partially ordered sets like that of collections of possibilities . The sum rule of probability theory is a direct consequence of the underlying lattice structure.…”

Section: Informationmentioning

confidence: 99%

The Physics of Information

Enßlin

Jasche

Skilling³

2019

Annalen der Physik

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InformationWhat is information? A short answer could be: Anything that changes our minds and states of thinking! Information is virtual. It can be carried by speech, an image, scratches on a stone, patterns in a photon field, the connections of neurons in our brains or even by the wavefunction of an electron. Information does not depend on the actual means of transmission nor does its identity change when changing the carrier. Information is physical. It needs to be sustained by a physical substrate and rearranging that substrate to contain the wanted information requires to spend energy or work. Without a physical world, information can not be stored, processed, or transmitted. Our world is permeated with different representations of information. Actually Information about the physical laws governing our Universe can be found everywhere and in everything.Could it then be that the real fundamental elements of this world are tiny bits of information? Could we get it from bit? [1] A number of researcher investigate this possibility, and several of their attempts to recover physical laws from the principles governing information are reported about in this special issue on physics of information. The articles by Gregg Jaeger, [2] by Ariel Caticha, [3] and by John Skilling and Kevin Knuth [4] address the reconstruction of quantum mechanics from information theory. Furthermore, Kevin Knuth and James Walsh [5] show how to recover space-time from particles sending each others signals in an initially geometry-free setting.A central element of most works on information is the concept of a probability. Probabilities can be defined via various routes. Most familiar might be their definitions by the frequentists school as relative frequencies of events (in the limit of infinite amounts of such). A more general definition of probabilities is the Bayesians' one, which regards probabilities as quantifiers of statements, expressing how much those should be expected to be true. Although this definition looks subjective on the first glance, it rests on a solid mathematical theorem by Richard Cox [6] that any extension of binary logic to a continuum of single valued truthfulness quantifier that respects a minimum of requirements must be isomorphic to probability theory. And this definition turns out to embrace the frequentists' definition in case that infinite numbers of comparable events are actually available and can be counted. The majority of authors of this special issues seem to favor the Bayesian perspective, which is probably a result of our own, the guest editors', preference and not necessarily a reflection of the prevailing position of contemporary scientists.The different views on probabilities of frequentists and Bayesians are addressed by the article of Allen Caldwell. [7] He builds a bridge between these antagonistic camps by explaining frequentists' constructions in a Bayesian language. In particular, his argumentation why the omnipresent p-value in statistical and experimental literature might often capture a releva...

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Lattices and Their Consistent Quantification

Cited by 8 publications

References 51 publications

The Symmetrical Foundation of Measure, Probability, and Quantum Theories

The Symmetrical Foundation of Measure, Probability, and Quantum Theories

Symmetries among Multivariate Information Measures Explored Using Möbius Operators

The Physics of Information

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