[Sorry for not answering earlier, I didn't find the inbox until recently.]
I perhaps was a bit unclear, but when I say "ideal bayesian" I mean a mathematical construct that does full bayesian updating i.e. incorporates all prior knowledge into its calculations. This is of course impossible for anyone not extremely ignorant of the world, which is why I called it a minor point.
An ideal bayesian calculation would include massive deductive work on e.g. the psychology of voting, knowledge of the functioning of this community in particular etc.
My comment wasn't really an objection. To do a full bayesian calculation of a real world problem is comparable to using quantum mechanics for macroscopic systems. One must use approximations; the hard part is knowing when they break down.
An information cascade is a problem in group rationality. Wikipedia has excellent introductions and links about the phenomenon, but here is a meta-ish example using likelihood ratios.
Suppose in some future version of this site, there are several well-known facts:
Let's talk about how the very first reader would vote. If they judged the post high quality, then they would multiply the prior likelihood ratio (6:4) times the bayes factor for a high private signal (4:1), get (6*4:4*1) = (6:1) and vote the post up. If they judged the post low quality then they would instead multiply by the bayes factor for a low private signal (1:4), get (6*1:4*4) = (3:8) and vote the post down.
There were two scenarios for the first reader (private information high or low). If we speculate that the first reader did in fact vote up, then there are two scenarios for the second scenario: There are two scenarios for the second reader:
Note that now there are two explanations for ending up two votes up. It could be that the second reader actually agreed, or it could be that the second reader was following the first reader and the prior against their personal judgement. That means that the third reader gets zero information from the second reader's personal judgement! The two scenarios for the third reader, and every future reader, are exactly analogous to the two scenarios for the second reader.
This has been a nightmare scenario of groupthink afflicting even diligent bayesians. Possible conclusions:
Note: Olle found an error that necessitated a rewrite. I apologize.