This is amazing, Kevin! Really cool and weird results.

I don't quite understand them, actually, in your first case with imperfect memories. In case it helps to ask: Suppose the fair coin lands heads 100 of the 200 times. Then Blues think it landed heads 100–101 times, and Reds think it landed heads 99–100 times, right? Is that the only difference we need enough to make their average credences diverge as widely as is shown? Or is there some cumulative effect of the memory lapses I'm not factoring in, suggested by the notion in your post title of lapses "adding up"? Are the Reds and Blues differing not just about the total number of heads, but about the number of heads in subsets of the flips?

Ah, good question! This is a bit unclear in hindsight, but what you say at the end is right: the "number of signals" axis refers to the number of *groups of 10 tosses* they update on. So each time-step involves tossing it 10 more times, and then conditioning on either "n heads out of 10" or "n or n+1 out of 10" heads, or "n-1 or n heads out of 10". So suppose a maximally biased-up (blue) and biased-down) (red) agent both see 5 of 10 tosses. Then the blue agent remembers "5 or 6 of 10", while the red ones remembers" 4 or 5 of 10". That counts as one round (or "signal"), so it's over the course of hundreds of those that the divergences start to be noticeable.

Ah, right, I see better now, I think. So in my case actually there would be 2000 coin flips, and if 1000 of them came up heads, then in the end Reds would distribute confidence between 2000 and 2200 heads, and Blues between 1800 and 2000, I think. Makes sense that they'd be pretty far apart by then.

This is amazing, Kevin! Really cool and weird results.

I don't quite understand them, actually, in your first case with imperfect memories. In case it helps to ask: Suppose the fair coin lands heads 100 of the 200 times. Then Blues think it landed heads 100–101 times, and Reds think it landed heads 99–100 times, right? Is that the only difference we need enough to make their average credences diverge as widely as is shown? Or is there some cumulative effect of the memory lapses I'm not factoring in, suggested by the notion in your post title of lapses "adding up"? Are the Reds and Blues differing not just about the total number of heads, but about the number of heads in subsets of the flips?

Ah, good question! This is a bit unclear in hindsight, but what you say at the end is right: the "number of signals" axis refers to the number of *groups of 10 tosses* they update on. So each time-step involves tossing it 10 more times, and then conditioning on either "n heads out of 10" or "n or n+1 out of 10" heads, or "n-1 or n heads out of 10". So suppose a maximally biased-up (blue) and biased-down) (red) agent both see 5 of 10 tosses. Then the blue agent remembers "5 or 6 of 10", while the red ones remembers" 4 or 5 of 10". That counts as one round (or "signal"), so it's over the course of hundreds of those that the divergences start to be noticeable.

Ah, right, I see better now, I think. So in my case actually there would be 2000 coin flips, and if 1000 of them came up heads, then in the end Reds would distribute confidence between 2000 and 2200 heads, and Blues between 1800 and 2000, I think. Makes sense that they'd be pretty far apart by then.