2012
DOI: 10.1109/tit.2011.2178139
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On Bregman Distances and Divergences of Probability Measures

Abstract: The paper introduces scaled Bregman distances of probability distributions which admit non-uniform contributions of observed events. They are introduced in a general form covering not only the distances of discrete and continuous stochastic observations, but also the distances of random processes and signals. It is shown that the scaled Bregman distances extend not only the classical ones studied in the previous literature, but also the information divergence and the related wider class of convex divergences o… Show more

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Cited by 46 publications
(53 citation statements)
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“…Another setting arises when higher dimensional aspects are compressed into real‐valued functionals, such as in the case of ϕ ‐divergences of probability measures P and Q . In a nutshell, a ϕ ‐divergence is of the formDitalicϕfalse(P,Qfalse)=ϕnormaldPnormaldQdQ,where ϕ is a convex function on [0, ∞) such that ϕ (1)=0; see, for example Stummer and Vajda () for a recent treatment and a delineation from Bregman distances. Any function ϕ of this form admits a mixture representation in terms of elementary functions or atoms kitalicθfalse(rfalse)=false|ritalicθfalse|1false(r1italicθ<r1false) for θ >0.…”
Section: Choquet Representationsmentioning
confidence: 99%
“…Another setting arises when higher dimensional aspects are compressed into real‐valued functionals, such as in the case of ϕ ‐divergences of probability measures P and Q . In a nutshell, a ϕ ‐divergence is of the formDitalicϕfalse(P,Qfalse)=ϕnormaldPnormaldQdQ,where ϕ is a convex function on [0, ∞) such that ϕ (1)=0; see, for example Stummer and Vajda () for a recent treatment and a delineation from Bregman distances. Any function ϕ of this form admits a mixture representation in terms of elementary functions or atoms kitalicθfalse(rfalse)=false|ritalicθfalse|1false(r1italicθ<r1false) for θ >0.…”
Section: Choquet Representationsmentioning
confidence: 99%
“…To embed the abovementioned two major divergence classes as a special case of Definition , let us first mention Stummer and Stummer and Vajda have shown that in case of ϕ (1) = 0 for the choice m ( y ): = q ( y ) ( yscriptY) the scaled Bregman distance becomes Bϕfalse(P,Q0.1emfalse‖0.1emQfalse)=Yqfalse(0.1emyfalse)·ϕ()pfalse(1ptyfalse)qfalse(1ptyfalse)0.5emnormaldλfalse(1ptyfalse)=:DϕCASfalse(P,Qfalse), which is nothing but the well‐known ϕ ‐divergence between P and Q (and on the right‐hand side, one has to additionally subtract ϕ (1) in case that it is not zero). The latter has been first studied by Csiszár and Ali and Silvey .…”
Section: The Divergence Frameworkmentioning
confidence: 99%
“…Scaled Bregman divergences, formally introduced by Stummer [24] and Stummer and Vajda [25], unify separable Bregman divergences [10] (defined below in Section 2.3) and f-divergences [8,9]. This paper uses scaled Bregman divergences as its basis, and accomplishes the following objectives:…”
Section: Goal Of This Papermentioning
confidence: 99%
“…Definition 2 (Notations) [25]: M denotes the space of all finite measures on a measurable space (X , A) and P ⊂ M the subspace of all probability measures. Unless otherwise explicitly stated P,R,M are mutually measure-theoretically equivalent measures on (X , A) dominated by a σ-finite measure λ on (X , A).…”
Section: Bregman Divergences and Scaled Bregman Divergencesmentioning
confidence: 99%
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