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Maximum Entropy Applied to Inductive Logic and Reasoning

by Jürgen Landes and Jon Williamson

Entropy 2015, 17(5), 3458-3460; https://0-doi-org.brum.beds.ac.uk/10.3390/e17053458 - 18 May 2015

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This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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Justifying Objective Bayesianism on Predicate Languages

by Jürgen Landes and Jon Williamson

Entropy 2015, 17(4), 2459-2543; https://0-doi-org.brum.beds.ac.uk/10.3390/e17042459 - 22 Apr 2015

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Abstract

Objective Bayesianism says that the strengths of one’s beliefs ought to be probabilities, calibrated to physical probabilities insofar as one has evidence of them, and otherwise sufficiently equivocal. These norms of belief are often explicated using the maximum entropy principle. In this paper [...] Read more.

Objective Bayesianism says that the strengths of one’s beliefs ought to be probabilities, calibrated to physical probabilities insofar as one has evidence of them, and otherwise sufficiently equivocal. These norms of belief are often explicated using the maximum entropy principle. In this paper we investigate the extent to which one can provide a unified justification of the objective Bayesian norms in the case in which the background language is a first-order predicate language, with a view to applying the resulting formalism to inductive logic. We show that the maximum entropy principle can be motivated largely in terms of minimising worst-case expected loss. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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Maximum Entropy and Probability Kinematics Constrained by Conditionals

by Stefan Lukits

Entropy 2015, 17(4), 1690-1700; https://0-doi-org.brum.beds.ac.uk/10.3390/e17041690 - 27 Mar 2015

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Abstract

Two open questions of inductive reasoning are solved: (1) does the principle of maximum entropy (PME) give a solution to the obverse Majerník problem; and (2) isWagner correct when he claims that Jeffrey’s updating principle (JUP) contradicts PME? Majerník shows that PME provides [...] Read more.

Two open questions of inductive reasoning are solved: (1) does the principle of maximum entropy (PME) give a solution to the obverse Majerník problem; and (2) isWagner correct when he claims that Jeffrey’s updating principle (JUP) contradicts PME? Majerník shows that PME provides unique and plausible marginal probabilities, given conditional probabilities. The obverse problem posed here is whether PME also provides such conditional probabilities, given certain marginal probabilities. The theorem developed to solve the obverse Majerník problem demonstrates that in the special case introduced by Wagner PME does not contradict JUP, but elegantly generalizes it and offers a more integrated approach to probability updating. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

211 KiB

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Maximum Relative Entropy Updating and the Value of Learning

by Patryk Dziurosz-Serafinowicz

Entropy 2015, 17(3), 1146-1164; https://0-doi-org.brum.beds.ac.uk/10.3390/e17031146 - 11 Mar 2015

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Abstract

We examine the possibility of justifying the principle of maximum relative entropy (MRE) considered as an updating rule by looking at the value of learning theorem established in classical decision theory. This theorem captures an intuitive requirement for learning: learning should lead to [...] Read more.

We examine the possibility of justifying the principle of maximum relative entropy (MRE) considered as an updating rule by looking at the value of learning theorem established in classical decision theory. This theorem captures an intuitive requirement for learning: learning should lead to new degrees of belief that are expected to be helpful and never harmful in making decisions. We call this requirement the value of learning. We consider the extent to which learning rules by MRE could satisfy this requirement and so could be a rational means for pursuing practical goals. First, by representing MRE updating as a conditioning model, we show that MRE satisfies the value of learning in cases where learning prompts a complete redistribution of one’s degrees of belief over a partition of propositions. Second, we show that the value of learning may not be generally satisfied by MRE updates in cases of updating on a change in one’s conditional degrees of belief. We explain that this is so because, contrary to what the value of learning requires, one’s prior degrees of belief might not be equal to the expectation of one’s posterior degrees of belief. This, in turn, points towards a more general moral: that the justification of MRE updating in terms of the value of learning may be sensitive to the context of a given learning experience. Moreover, this lends support to the idea that MRE is not a universal nor mechanical updating rule, but rather a rule whose application and justification may be context-sensitive. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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Relational Probabilistic Conditionals and Their Instantiations under Maximum Entropy Semantics for First-Order Knowledge Bases

by Christoph Beierle, Marc Finthammer and Gabriele Kern-Isberner

Entropy 2015, 17(2), 852-865; https://0-doi-org.brum.beds.ac.uk/10.3390/e17020852 - 13 Feb 2015

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Abstract

For conditional probabilistic knowledge bases with conditionals based on propositional logic, the principle of maximum entropy (ME) is well-established, determining a unique model inductively completing the explicitly given knowledge. On the other hand, there is no general agreement on how to extend the [...] Read more.

For conditional probabilistic knowledge bases with conditionals based on propositional logic, the principle of maximum entropy (ME) is well-established, determining a unique model inductively completing the explicitly given knowledge. On the other hand, there is no general agreement on how to extend the ME principle to relational conditionals containing free variables. In this paper, we focus on two approaches to ME semantics that have been developed for first-order knowledge bases: aggregating semantics and a grounding semantics. Since they use different variants of conditionals, we define the logic PCI, which covers both approaches as special cases and provides a framework where the effects of both approaches can be studied in detail. While the ME models under PCI-grounding and PCI-aggregating semantics are different in general, we point out that parametric uniformity of a knowledge base ensures that both semantics coincide. Using some concrete knowledge bases, we illustrate the differences and common features of both approaches, looking in particular at the ground instances of the given conditionals. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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A Foundational Approach to Generalising the Maximum Entropy Inference Process to the Multi-Agent Context

by George Wilmers

Entropy 2015, 17(2), 594-645; https://0-doi-org.brum.beds.ac.uk/10.3390/e17020594 - 02 Feb 2015

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Abstract

The present paper seeks to establish a logical foundation for studying axiomatically multi-agent probabilistic reasoning over a discrete space of outcomes. We study the notion of a social inference process which generalises the concept of an inference process for a single agent which [...] Read more.

The present paper seeks to establish a logical foundation for studying axiomatically multi-agent probabilistic reasoning over a discrete space of outcomes. We study the notion of a social inference process which generalises the concept of an inference process for a single agent which was used by Paris and Vencovská to characterise axiomatically the method of maximum entropy inference. Axioms for a social inference process are introduced and discussed, and a particular social inference process called the Social Entropy Process, or SEP, is defined which satisfies these axioms. SEP is justified heuristically by an information theoretic argument, and incorporates both the maximum entropy inference process for a single agent and the multi–agent normalised geometric mean pooling operator. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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The Information Geometry of Bregman Divergences and Some Applications in Multi-Expert Reasoning

by Martin Adamčík

Entropy 2014, 16(12), 6338-6381; https://0-doi-org.brum.beds.ac.uk/10.3390/e16126338 - 01 Dec 2014

Cited by 16 | Viewed by 9101

Abstract

The aim of this paper is to develop a comprehensive study of the geometry involved in combining Bregman divergences with pooling operators over closed convex sets in a discrete probabilistic space. A particular connection we develop leads to an iterative procedure, which is [...] Read more.

The aim of this paper is to develop a comprehensive study of the geometry involved in combining Bregman divergences with pooling operators over closed convex sets in a discrete probabilistic space. A particular connection we develop leads to an iterative procedure, which is similar to the alternating projection procedure by Csiszár and Tusnády. Although such iterative procedures are well studied over much more general spaces than the one we consider, only a few authors have investigated combining projections with pooling operators. We aspire to achieve here a comprehensive study of such a combination. Besides, pooling operators combining the opinions of several rational experts allows us to discuss possible applications in multi-expert reasoning. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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What You See Is What You Get

by Jeff B. Paris

Entropy 2014, 16(11), 6186-6194; https://0-doi-org.brum.beds.ac.uk/10.3390/e16116186 - 21 Nov 2014

Cited by 18 | Viewed by 5222

Abstract

This paper corrects three widely held misunderstandings about Maxent when used in common sense reasoning: That it is language dependent; That it produces objective facts; That it subsumes, and so is at least as untenable as, the paradox-ridden Principle of Insufficient Reason. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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