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.
Yes, meta-probabilities are probabilities, although somewhat odd ones; they obey the normal rules of probability. Jaynes discusses this in his Chapter 18; his discussion there is worth a read.
The statement “probability estimates are not, by themselves, adequate to make rational decisions” was meant to describe the entire sequence, not this article.
I’ve revised the first paragraph of the article, since it seems to have misled many readers. I hope the point is clearer now!
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.
Thanks, that’s really funny! “On the other hand” is my general approach to life, so I’m happy to argue with myself.
And yes, I’m steelmanning. I think this approach is an excellent one in some cases; it will break down in others. I’ll present a first one in the next article. It’s another box you can put coins in that (I’ll claim) can’t usefully be modeled in this way.
Here’s the quote from Jaynes, by the way:
What are we doing here? It seems almost as if we are talking about the ‘probability of a probability’.
Pending a better understanding of what that means, let us adopt a cautious notation that will avoid giving possibly wrong impressions. We are not claiming that P(Ap|E) is a ‘real probability’ in the sense that we have been using that term; it is only a number which is to obey the mathematical rules of probability theory.
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.
Yes, meta-probabilities are probabilities, although somewhat odd ones; they obey the normal rules of probability. Jaynes discusses this in his Chapter 18; his discussion there is worth a read.
The statement “probability estimates are not, by themselves, adequate to make rational decisions” was meant to describe the entire sequence, not this article.
I’ve revised the first paragraph of the article, since it seems to have misled many readers. I hope the point is clearer now!
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.
Thanks, that’s really funny! “On the other hand” is my general approach to life, so I’m happy to argue with myself.
And yes, I’m steelmanning. I think this approach is an excellent one in some cases; it will break down in others. I’ll present a first one in the next article. It’s another box you can put coins in that (I’ll claim) can’t usefully be modeled in this way.
Here’s the quote from Jaynes, by the way: