Probability and time

Probabilistic reasoning is often attributed a temporal meaning, in which conditioning is regarded as a normative rule to compute future beliefs out of current beliefs and observations. However, the well-established ‘updating interpretation’ of conditioning is not concerned with beliefs that evolve in time, and in particular with future beliefs. On the other hand, a temporal justification of conditioning was proposed already by De Moivre and Bayes, by requiring that current and future beliefs be consistent. We reconsider the latter approach while dealing with a generalised version of the problem, using a behavioural theory of imprecise probability in the form of coherent lower previsions as well as of coherent sets of desirable gambles, and letting the possibility space be finite or infinite. We obtain that using conditioning is normative, in the imprecise case, only if one establishes future behavioural commitments at the same time of current beliefs. In this case it is also normative that present beliefs be conglomerable, which is a result that touches on a long-term controversy at the foundations of probability. In the remaining case, where one commits to some future behaviour after establishing present beliefs, we characterise the several possibilities to define consistent future assessments; this shows in particular that temporal consistency does not preclude changes of mind. And yet, our analysis does not support that rationality requires consistency in general, even though pursuing consistency makes sense and is useful, at least as a way to guide and evaluate the assessment process. These considerations narrow down in the special case of precise probability, because this formalism cannot distinguish the two different situations illustrated above: it turns out that the only consistent rule is conditioning and moreover that it is not rational to be willing to stick to precise probability while using a rule different from conditioning to compute future beliefs; rationality requires in addition the disintegrability of the present-time probability.