Thursday, February 23, 2012

Infinite lotteries and infinitesimal probabilities

The argument in this post is based on a construction by Dubins (see Example 2.1 here) that I've switched into an infinitesimal case.

Suppose you can have an infinite lottery with ticket numbers 1,2,3,... and each ticket has infinitesimal probability (perhaps the same one for each). Then really weird stuff can happen. Say I toss a fair coin, but don't show you the result. Instead, you know for sure that I will do this:

  • If the coin was tails, I run an infinite lottery with ticket numbers 1,2,3,... and with each ticket having infinitesimal probability
  • If the coin was heads, I run an infinite lottery with the same ticket numbers, but now the probability of ticket n is 2n.
And you know for sure that I will then announce the result of the lottery.

Here's the oddity. No matter what my announcement, you will end up all but certain—i.e., assigning a probability infinitesimally short of 1—that the coin was heads. Here's why. Suppose I announce ticket n. Now, P(n|heads)=2n but P(n|tails) is infinitesimal. Plugging these facts into Bayes' theorem, and assuming that your prior probability for heads was 1/2 (actually, all that's needed is that it be neither zero nor infinitesimal), your posterior probability P(heads|n) ends up equal to 1−a where a is infinitesimal.

So I can rationally force you to be all but certain that it was heads, simply by telling you the result of my lottery experiment. And by reversing the arrangement, I could force you to be all but certain that it was tails. Thus there is something pathological about the infinite lottery with infinitesimal probabilities.

This is, to me, yet another of the somewhat unhappy results that show that probability theory has a quite limited sphere of epistemological application.

6 comments:

Jonah Schupbach said...

Help me out. I don't see why examples like this should convince us that the epistemic import of the probability calculus is quite limited. I mean, what this tells me is that I better think twice before applying the probability theory as an epistemic theory in scenarios involving the infinite. But do you think this is severely going to limit the probability theory as an epistemic theory??

Alexander R Pruss said...

Well, whether a limit is severe depends on what our interests are or should be.

Our epistemic interests include or should include things like fine-tuning, the problem of evil, whether skepticism is true, multiple universe theories, Boltzmann brains, the lawfulness of the universe, why there is something rather than nothing, etc. In all of these cases, we are working at or beyond the boundary of the epistemological applicability of probability theory.

Now, if our interests were only in things like whether some drug reduces the risk of cancer, whether one rather than another theory of stellar formation is true, etc., then probabilistic reasoning could be just great.

I don't want to say that the latter kinds of things aren't important. They are often big questions, but they aren't The Big Questions. :-)

Jonah Schupbach said...

Do you think that these are all cases in which the probability theory loses its epistemic import on account of the sort of issues you raise in this post (i.e., on account of counterexamples involving infinity)? I.e., you list fine-tuning, the problem of evil, whether skepticism is true, multiple universe theories, Boltzmann brains, the lawfulness of the universe, and why there is something rather than nothing all as issues of epistemic interest (of course, I agree!) but which lie beyond the reach of a probabilistic epistemology. Is that because all of these cases involve infinity in troublesome ways? Or do you have other counterexamples to a probability-informed epistemology in mind with regards to some of these?

Alexander R Pruss said...

Good question. I think all of these cases are difficult to handle with probabilistic methods because either (a) some sort infinity-based problems show up (not necessarily the one I mention in this post) or (b) we are dealing with spaces of possibilities that there just is no natural probability measure on (e.g., we're not going to get a natural probability measure on the space of all laws for much the reason we aren't going to get a natural probability measure on the space of, say, all continuous functions) or both. Actually, I think (b) may be the more fundamental issue, and (a) just a symptom.

Alexander R Pruss said...

Bibliographic note: My paper building on this stuff has been accepted by Thought.

Alexander R Pruss said...

PDF is here.