On a posterior information process for parametric families of experiments

In a filtered statistical experiment a priori and a posteriori probability measures are defined on an abstract parametric space. The information in the posterior, given the prior, is defined by the usual Kullback-Leibler formula. Certain properties of this quantity is investigated in the context of so-called arithmetic and geometric measures and arithmetic and geometric processes. Interesting multiplicative decompositions are presented that involve Hellinger processes indexed both by prior and by posterior distributions.