Ergodic theorems for lower 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 observations are collected, $\mathcal{P}$ is updated using Jeffrey's rule of conditioning, a generalization of 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 give a rule, that we call Jeffrey-Geometric rule, to update lower probabilities associated with the elements of $(\mathcal{P}^*_k)$.