Suppose you update your credences via Bayesian update at discrete moments of time (i.e., at any future time, your credence is the result of a finite number of Bayesian updates from your present credence). Then it can be proved that you cannot be sure (i.e., assign probability one) that your credence will ever be higher than it is now, and similarly you cannot be sure that your credence will ever be lower than it is now.
The same is not true for continuous Bayesian update, as is shown by Alan Hajek’s Cable Guy story. Cable Guy will come tomorrow between 8:00 am and 4:00 pm, with 4:00 pm included but 8:00 am excluded. Your current credence that they will come in the morning is 1/2 and your current credence that they will come in the afternoon is also 1/2.
Then it is guaranteed that there will be a time after 8:00 am when Cable Guy hasn’t come yet. At that time, because you have ruled out some of the morning possibilities but none of the afternoon possibilities, your credence that the Cable Guy will come in the afternoon will have increased and your credence that the Cable Guy will come in the morning will have decreased.
Proof of fact in first paragraph: A Bayesian agent’s credences are a martingale. To obtain a contradiction, suppose there is probability 1 that the credences will go above their current value. Let Cn be the agent’s credence after the nth update, and consider everything from the point of view of the agent right now, before the updates, with current credence r. Let τ be the first time such that Cτ > r (this is defined with probability one). By Doob’s Optional Sampling Theorem, E[Cτ] = r. But this contradicts the inequality Cτ > r.
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