The impression I get of the difference here between a “belief” and a “hypothesis” is something like this:

I have the belief that the sun will continue to rise for a long long time.

This is probably “true.”

I have the hypothesis that the sun will rise tomorrow morning with probability .999999

Conservation of expected evidence requires that in pure Bayesian fashion, if it does rise tomorrow my probability will rise to .9999991 and if it doesn’t it will shoot down to .3 or something in a way that makes the view of all possible shifts a random walk.

That is, if your hypothesis is “true” you have great confidence, if it is true too often you are underconfident and the hypothesis has an issue.

By “belief”, grandparent means probability of a hypothesis, which does bob around without trend in a perfect Bayesian reasoner.

The impression I get of the difference here between a “belief” and a “hypothesis” is something like this:

I have the belief that the sun will continue to rise for a long long time.

This is probably “true.”

I have the hypothesis that the sun will rise tomorrow morning with probability .999999

Conservation of expected evidence requires that in pure Bayesian fashion, if it does rise tomorrow my probability will rise to .9999991 and if it doesn’t it will shoot down to .3 or something in a way that makes the view of all possible shifts a random walk.

That is, if your hypothesis is “true” you have great confidence, if it is true too often you are underconfident and the hypothesis has an issue.