Regret and Jeffreys Integrals in Exp. Families

The problem of whether minimax redundancy, minimax regret and Jeffreys integrals are finite or infinite are discussed

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 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 a