α-Parallel prior and its properties

It is known that the Jeffreys prior plays an important role in statistical inference. In this paper, we generalize the Jeffreys prior from the point of view of information geometry and introduce a one-parameter family of prior distributions, which we named the α-parallel priors. The α-parallel prior is defined as the parallel volume element with respect to the α-connection and coincides with the Jeffreys prior when α = 0. Further, we analyze asymptotic behavior of the various estimators such as the projected Bayes estimator (the estimator obtained by projecting the Bayes predictive density onto the original class of distributions) and the minimum description length (MDL) estimator, when the α-parallel prior is used. The difference of these estimators from maximum-likelihood estimator (MLE) due to the α-prior is shown to be regulated by an invariant vector field of the statistical model. Although the Jeffreys prior always exists, the existence of α-parallel prior with α ≠ 0 is not always guaranteed. Hence, we consider conditions for the existence of the a-parallel prior, elucidating the conjugate symmetry in a statistical model.