Hi, I have a site tech question. (Sorry if this is the wrong place to post that!—I couldn't find any other.)

I can't find a way to get email notifications of comment replies (i.e. when my inbox icon goes red). If there is one, how do I turn it on?

If there isn't one, is that a deliberate design feature, or a limitation of the software, or...?

Thanks (and thanks especially to whoever does the system maintenance here—it must be a big job.)

We could call this meta-probability, although that’s not a standard term.

Then why use it instead of learning the standard terms and using those? This might sound like pedantic, but it matters because this kind of thing leads to proliferation of unnecessary jargon and sometimes reinventing the wheel.

Are we talking about conditional probability? Joint probability?

Also, a minor nitpick about your next-to-last figure: given what's said about the boxes, it's not two bell curves centered at 0 and 0.9. It should be a point mass (vertical line) at 0 and a bell curve centered at 0.9.

If you enjoy this sort of thing, you might like to work out what the exact optimal algorithm is.

I guess this is a joke. From wikipedia: "Originally considered by Allied scientists in World War II, it proved so intractable that, according to Peter Whittle, it was proposed the problem be dropped over Germany so that German scientists could also waste their time on it.[10]" (note that your wikipedia-link is broken)

FWIW, I understood David to be requesting some specific examples of how members of the set "everyone else ever" handle this problem, which on your account is the same as how Jaynes handles it, in order to more clearly see the similarity you reference.

The “meta-probability” approach I’ve taken here is the Ap distribution of E. T. Jaynes. I find it highly intuitive, but it seems to have had almost no influence or application in practice. We’ll see later that it has some problems, which might explain this.

I don't see how this differs from how anyone else ever handles this problem. I hope you explain the difference in this example, before going on to other examples.

I'm looking forward to the rest of your sequence, thanks!

I was recently reading through a month-old blog post where one lousy comment was arguing against a strawman of Bayesian reasoning wherein you deal with probabilities by "mushing them all into a single number". I immediately recollected that the latest thing I saw on LessWrong was a fantastic summary of how you can treat mixed uncertainty as a probability-distribution-of-probability-distributions. I considered posting a belated link in reply, until I discovered that the lousy comment was written by David Chapman and the fantastic summary was written by David_Chapman.

I'm not sure if later you're going to go off the rails or change my mind or what, but so far this looks like one of the greatest attempts at "steelmanning" that I've ever seen on the internet.

The statement "probability estimates are not, by themselves, adequate to make rational decisions" could apparently have been replaced with the statement "my definition of the phrase 'probability estimates' is less inclusive than yours" - what you call a "meta-probability" I would have just called a "probability". In a world where both epistemic and aleatory uncertainty exist, your expectation of events in that world is going to look like a probability distribution over a space of probability distributions over outputs; this is still a probability distribution, just a much more expensive one to do approximate calculations with.

I'm not sure I follow this. There is no prior distribution for the per-coin payout probabilities that can accurately reflect all our knowledge.

Are we talking about the Laplace vs. fair coins? Are you claiming there's no prior distribution over sequences which reflects our knowledge? If so I think you are wrong as a matter of math.