Conditional models: Coherence and inference through sequences of joint mass functions

Abstract

We call a conditional model any set of statements made of conditional probabilities or expectations. We take conditional models as primitive compared to unconditional probability, in the sense that conditional statements do not need to be derived from an unconditional probability. We focus on two problems: ( coherence) giving conditions to guarantee that a conditional model is self-consistent; ( inference) delivering methods to derive new probabilistic statements from a self-consistent conditional model. We address these problems in the case where the probabilistic statements can be specified imprecisely through sets of probabilities, while restricting the attention to finite spaces of possibilities. Using Walley's theory of coherent lower previsions, we fully characterise the question of coherence, and specialise it for the case of precisely specified probabilities, which is the most common case addressed in the literature. This shows that coherent conditional models are equivalent to sequences of (possibly sets of) unconditional mass functions. In turn, this implies that the inferences from a conditional model are the limits of the conditional inferences obtained by applying Bayes’ rule, when possible, to the elements of the sequence. In doing so, we unveil the tight connection between conditional models and zero-probability events.