Ergodic theorems for imprecise probability kinematics
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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 P of plausible prior probability measures; as more and more data are collected, 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 (P^*_k) of successive updates of P and we develop an ergodic theory for its limit, for countable and uncountable sample space Ω. A result of this ergodic theory is a strong law of large numbers when Ω is uncountable. We also develop procedure for updating lower probabilities using Jeffrey's rule of conditioning.
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