Vanessa Kosoy's Shortform

Vanessa Kosoy

This is a special post for quick takes by Vanessa Kosoy. Only they can create top-level comments. Comments here also appear on the Quick Takes page and All Posts page.

I have repeatedly argued for a departure from pure Bayesianism that I call "quasi-Bayesianism". But, coming from a LessWrong-ish background, it might be hard to wrap your head around the fact Bayesianism is somehow deficient. So, here's another way to understand it, using Bayesianism's own favorite trick: Dutch booking!

Consider a Bayesian agent Alice. Since Alice is Bayesian, ey never randomize: ey just follow a Bayes-optimal policy for eir prior, and such a policy can always be chosen to be deterministic. Moreover, Alice always accepts a bet if ey can choose which side of the bet to take: indeed, at least one side of any bet has non-negative expected utility. Now, Alice meets Omega. Omega is very smart so ey know more than Alice and moreover ey can predict Alice. Omega offers Alice a series of bets. The bets are specifically chosen by Omega s.t. Alice would pick the wrong side of each one. Alice takes the bets and loses, indefinitely. Alice cannot escape eir predicament: ey might know, in some sense, that Omega is cheating em, but there is no way within the Bayesian paradigm to justify turning down the bets.

A possible counterargument is, we don't need to depart far from Bayesianism to win here. We only need to somehow justify randomization, perhaps by something like infinitesimal random perturbations of the belief state (like with reflective oracles). But, in a way, this is exactly what quasi-Bayesianism does: a quasi-Bayes-optimal policy is in particular Bayes-optimal when the prior is taken to be in Nash equilibrium of the associated zero-sum game. However, Bayes-optimality underspecifies the policy: not every optimal reply to a Nash equilibrium is a Nash equilibrium.

This argument is not entirely novel: it is just a special case of an environment that the agent cannot simulate, which is the original motivation for quasi-Bayesianism. In some sense, any Bayesian agent is dogmatic: it dogmatically beliefs that the environment is computationally simple, since it cannot consider a hypothesis which is not. Here, Omega exploits this false dogmatic belief.

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I'm not sure I understand the argument here correctly. It seems like the intended argument is something like this: "Omega has access to an infinite number of fair coinflips. Alice can do no better than guess, and Alice cannot guess every coin-flip correctly. Omega knows how Alice will guess, and also knows how each coinflip will land. Therefore, Omega can choose to ask Alice about only the coinflips Alice will guess incorrectly (of which there will be at least one). Alice therefore surely loses money from bets placed." This argument uses the assumption that Alice can't change eir beliefs in response to learning that Omega has proposed specific bets and not others. This might seem concerning, because it seems like precisely what Alice should do, if Alice understands the situation: Alice should expect to lose any bet proposed by Omega. However, this assumption is perfectly normal for Dutch Book arguments. Such an objection would rule out all the usual Dutch Books. I think the classic Dutch Book arguments in fact illustrate a useful idea, even with this 'flaw', so I allow it. More concerningly, the argument assumes Omega has knowledge of how the coins will land. This is a significant departure from classical Dutch Books. It seems clear that a bookie can reliably make money from gamblers if the bookie knows which horse will win which race; this is not, in the classical way of thinking, a testament to the irrationality of the gamblers. It appears to me that this is all that is happening in the above argument. A second quibble is that in classical Dutch Book arguments, the bookie will surely make money. In the argument above, the bookie only almost surely makes money: since Omega relies on Alice making a bad guess, Omega makes money with probability 1, but not with (logical) certainty. Considering these two violations of the pre-existing norms of Dutch Books, what should we make of the proposed Dutch Book argument? It intuitively makes sense to me that Infrabayes mig

This argument uses the assumption that Alice can't change eir beliefs in response to learning that Omega has proposed specific bets and not others.

Not true. Changing her beliefs in response to Omega's proposal doesn't help her. Imagine that Alice is given a choice between

Take a bet that pays +2 if X and -1 if not-X.
Take a bet that pays +2 if not-X and -1 if X.
Refuse both bets.

No matter what probability Alice assigns to X after her update, "normal" Bayesian calculus (really CDT calculus, see below) mandates that she chooses 1 or 2, not 3.

It seems clear that... (read more)

Vanessa Kosoy's Shortform

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Vanessa Kosoy's Shortform

6