This is a special post for quick takes by Richard_Ngo. Only they can create top-level comments. Comments here also appear on the Quick Takes page and All Posts page.
(I expect that Scott, Abram or some others have already pointed this out, but somehow this clicked for me only recently. Pointers to existing discussions appreciated.)
A Bayesian update can be seen as a special case of a prediction market resolution.
Specifically, a Bayesian update is the case where each "hypothesis" has bet all their wealth across some combination of outcomes, and then the pot is winner-takes-all (or split proportionally when there are multiple winners).
The problem with Bayesianism is then obvious: what happens when there are no winners? Your epistemology is "bankrupt", the money vanishes into the ether, and bets on future propositions are undefined.
So why would a hypothesis go all-in like that? Well, that's actually the correct "cooperative" strategy in a setting where you're certain that at least one of them is exactly correct.
To generalize Bayesianism, we want to instead talk about what the right "cooperative" strategy is when a) you don't think any of them are exactly correct, and b) when each hypothesis has goals too, not just beliefs.
Yep, see this paper - Bayesian updating is the same as having all the hypotheses in your head be agents in a prediction market using the Kelly criterion, where their inital wealths are your prior probabilities on those hypotheses, the market price of some observation is your marginal probability over that observation, and the wealths after the bets resolve are the posterior distributions.
3Cole Wyeth
Indeed in algorithmic information theory, the lower semicomputable semimeasures are an example of "subprobabilities." Much has been written about updating in this context.