Bayesian inference under ambiguity: Conditional prior belief functions

Bayesian inference under imprecise prior information is studied: the starting point is a precise strategy σ and a full B-conditional prior belief function BelB, conveying ambiguity in probabilistic prior information. In finite spaces, we give a closed form expression for the lower envelope P of the class of full conditional probabilities dominating BelB, σ and, in particular, for the related “posterior probabilities”. The assessment BelB, σ is a coherent lower conditional probability in the sense of Williams and the characterized lower envelope P coincides with its natural extension.