Ergodic theorems for imprecise probability kinematics

In a standard Bayesian setting, there is often ambiguity in prior choice, as one may have not sufficient information to uniquely identify a suitable prior probability measure encapsulating initial beliefs. To overcome this, we specify a set $\mathcal{P}$ of plausible prior probability measures; as more and more data are collected, $\mathcal{P}$ is updated using Jeffrey's rule of conditioning, an alternative to Bayesian updating which proves to be more philosophically compelling in many situations. We build the sequence $(\mathcal{P}^*_k)$ of successive updates of $\mathcal{P}$ and we provide an ergodic theory to analyze its limit, for both countable and uncountable sample spaces. A result of this ergodic theory is a strong law of large numbers in the uncountable setting. We also develop a procedure for updating lower probabilities using Jeffrey's rule of conditioning.