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Finiteness of Redundancy, Regret, *Shtarkov* Sums, and Jeffreys *Integrals* in Exponential Families

, 2009

"... The normalized maximum likelihood (NML) distribution plays a fundamental role in the MDL approach to statistical inference. It is only defined for statistical families with a finite Shtarkov sum. Here we characterize, for 1-dimensional exponential families, when the Shtarkov sum is finite. This tur ..."

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The normalized maximum likelihood (NML) distribution plays a fundamental role in the MDL approach to statistical inference. It is only defined for statistical families with a finite

*Shtarkov*sum. Here we characterize, for 1-dimensional exponential families, when the*Shtarkov*sum is finite###
Regret and Jeffreys *Integrals* in Exp. Families

"... Let fP j 2 can g be a 1-dimensional exponential family given in a canonical parameterization, dP dQ 1 Z ( ) e x; (1) where Z is the partition function Z ( ) = R exp ( x) dQx, and can: = f j Z ( ) < 1g is the canonical parameter space. We let sup = supf j 2 can g, and inf likewise. The elements o ..."

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of the exponential family are also parametrized by their mean value. We write for the mean value corresponding to the canonical parameter and for the canonical parameter corresponding to the mean value: For any x the maximum likelihood distribution is P x: The

*Shtarkov**integral*S is de ned as### unknown title

, 903

"... Let {Pβ | β ∈ Γ can} be a 1-dimensional exponential family given in a canonical parameterization, dPβ dQ 1 Z(β) eβx, (1) where Z is the partition function Z(β) = ∫ exp(βx) dQx, and Γcan: = {β | Z(β) < ∞} is the canonical parameter space. We let βsup = sup{β | β ∈ Γcan}, and βinf likewise. The e ..."

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. The elements of the exponential family are also parametrized by their mean value µ. We write µβ for the mean value corresponding to the canonical parameter β and βµ for the canonical parameter corresponding to the mean value µ. For any x the maximum likelihood distribution is Pβx. The

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