Methods of estimating distances between members of (r, r) exponential families are considered. The first replaces the parameters in the geodesic distance associated with the information metric by their maximum likelihood estimates. The second is based on the family of predictive densities corresponding to Jeffrey’s invariant prior, using the sufficient statistics as co-ordinates of a Riemannian manifold. In all examples considered, the resulting estimative and predictive distances differ in form by only a simple multiple, the predictive distance being the shorter, and interesting geometrical relationships associated with flatness are also observed. Finally, the effect of the conjugate priors on distances and flatness is considered.
KeywordsRao’s distance (r, r) exponential families Jeffreys’ invariant prior Predictive density Flatness Conjugate priors
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