Learning under Prior Ignorance

We study the problem of learning the probability distribution of a multinomial variable from an observed sequence of signals, starting in a condition of ignorance about this distribution. We show that not all signals are suited for producing non-vacuous inferences under prior ignorance. To discriminate between informative and non-informative signals, we derive a general necessary condition for learning under prior ignorance. We concentrate then on the important case of multinomial signals and derive necessary and sufficient conditions for learning under prior ignorance, extending thus the imprecise Dirichlet model of Walley (1996) to general multinomial signals. Finally, an alternative model that produces non-vacuous inferences for any multinomial signal is discussed.