Let's say we want to know if all ravens are black, in a more direct way than just googling it. Formally, we want to know whether this is true: if x is a raven, x is black. Equivalently: if x is not black, then x is not a raven. If seeing a black raven makes us more likely to believe this statement (and it should, by conservation of expected probability), then so should seeing a non-black, non-raven, because those statements are logically equivalent and the information comes from the same use of the if-then format.

But seeing a white sheet of paper really doesn't have anything to do with ravens. So this is sort of bananas. That's the Raven Paradox. It's the Peter Griffin-esque "a black raven is a raven, but a mystery box could be anything! It could even be a non-black raven! You know how much I've wanted to find one of those!" approach to testing a hypothesis. There's something going on there, it's just not quite normal.

The classic Bayesian resolution to this confused idea is a bit weak. I.J. Good's argument indicates this information is exceptionally weak, proportional to r/(2N - 2b), where N is the number of things, b is the number of black things, and r is the number of ravens. Since ravens are a small percentage of "things", this number is exceptionally minimal.

But how many "things" are there, really? There are clearly at least particles, which gets us up to at least 1080. But things have differently colored components, and things span all possible sizes (that's right, there are non-particle "things" too). I'm wearing a blue shirt, but that blue shirt contains several grey buttons, and many varied-in-color threads. A better estimate would be the power set's size, giving 210^80 possible things, an unimaginably huge number.

But if we're concerned about the 2*(non-black things) denominator, it's even larger. Most things don't even have a color. "Modern Accounting Practices" might not reflect light, but I think we'd be hard pressed to describe it as black. It's an idea (and very much a thing) – and the set of ideas are a countable infinity.

But, of course, there's only a few ravens, really. Millions, at most? So the Bayesian update is of size zero (in log-odds form, obvious, you'd multiply by 1). There's no information content, in a precise, mathematical sense. I'm not entirely sure why the classic Bayesian discussion of this leaves that fact out.

I could, of course, comb through the alternate Bayesian discussions, but why? Clearly they wouldn't come to the conclusion I ought to be actually convinced about ravens by a ream of paper. I'd anticipate they'd only speak up about their methodology when it leads to non-catastrophically-flawed conclusions. Reviewing the literature of people agreeing in a complex way is really quite boring, although I wouldn't fault you for the effort (and if someone mails in a citation for the formal version of this argument, I'd definitely immortalize them in this largely-unread blog).