Bounding changes in probability over time : It is unlikely that you will change your mind very much very often

Under some circumstances, a probabilistic reasoner who encounters a sequence of observations will change his estimate of the likelihood of a proposition φ from very low to very high and back again many times. However, the prior probability of such a sequence of observations is necessarily very small. We compute an exact value of the maximal value of this probability, as a function of the amount of change and the number of changes. The calculation does not involve any independence assumptions over the observations.