Suppose a coin lands heads 75% of the time, but Bob doesn't know this.
Bob initially believes the coin is equally likely to be heads-biased (75% heads, which is the actual state) or tails-biased (75% tails, which is not the actual state).
Bob flips the coin over and over again, and he updates his beliefs using Bayes's rule. So if the first coin lands heads, and he updates his beliefs to 75% chance of the coin being heads-based. If the second flip lands heads, he'll update to 90%. If the second flip is tails, he updates back to 50%. Call the absolute movement the sum of the absolute values of the difference in beliefs after each flip.
I'm curious about the expected absolute movement in Bob's beliefs about which bias the coin has. Suppose he flips coins forever. Of course we know eventually he tends to get close to believing the coin is 100% likely to be heads-biased--with the proper quantifiers and limits of course--so the total expected movement is .5. But what is the expected absolute movement from this infinite sequence of belief movements? In particular, is it finite?