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 10^{80}. 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 2^{10^80} 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).