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 $Bel_B$, conveying ambiguity in probabilistic prior information. In finite spaces, we give a closed form expression for the lower envelope $\underlineP$ of the class of full conditional probabilities dominating $(Bel_B,σ)$ and, in particular, for the related “posterior probabilities”. The assessment $(Bel_B,σ)$ is a coherent lower conditional probability in the sense of Williams and the characterized lower envelope $\underlineP$ coincides with its natural extension.