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There's no paradox here. Imagine that in the world of things, about 1 in 100 are black and about 1 in 1,000,000 are ravens. It's true as far as it goes that obs

There's no paradox here. Imagine that in the world of things, about 1 in 100 are black and about 1 in 1,000,000 are ravens. It's true as far as it goes that observing a black raven and observing a nonblack nonraven are both consistent with (and thus evidence for) the proposition that all ravens are black, but the first observation is about 10,000 times as strong of evidence as the second. Because Bayes.

deleted 12y ago
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andrewmutz 12y ago
Also, if the world of things is infinite, then this doesn't hold up, right? if the number of things is infinite then observations of nonblack nonravens is not evidence of the proposition, right?
- xxxyy 12y ago
  It still does hold up, but becomes more complicated. In general, there is no uniform distribution over any infinite set. We might: a) think about a pipe spitting out objects that have two possible features - being a raven and being black, or b) use a nonuniform distribution over an infinite set of objects and integrate (sum) to get a similar Bayesian result.
  - x0x0 12y ago
    What do you mean there is no uniform distribution over any infinite set? There is the uniform distribution over [0,1] which is both infinite and not even countable. I'm confused.
    - xxxyy 12y ago
      I meant a distribution with "discrete" probability, i.e. a distribution where the probabilities of singletons are all equal and nonzero, so that a simple Bayesian argument could possibly be extended. My bad for not being precise enough. Perhaps I should have stuck to natural numbers in my previous comment, otherwise yes, you can have uniform distributions with respect to some additional structure of the probability space (like [0,1] with the Lebesgue measure you suggest).
brownbat 12y ago
I like that there are 11 distinct Bayesian solutions to the problem, at least one of them claims it is "more Bayesian" than the other Bayesian solutions. I'd like to know more about that particular solution. Unfortunately the volume containing the essay retails for around $250. I'm not sure I want to know that badly... http://www.barnesandnoble.com/w/essays-in-honor-of-carl-g-hempel-nicholas-rescher/1101630822?ean=9789027700858 http://www.barnesandnoble.com/w/essays-in-honor-of-carl-g-he...

deleted 12y ago

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andrewmutz 12y ago

Also, if the world of things is infinite, then this doesn't hold up, right? if the number of things is infinite then observations of nonblack nonravens is not evidence of the proposition, right?

xxxyy 12y ago
It still does hold up, but becomes more complicated. In general, there is no uniform distribution over any infinite set. We might: a) think about a pipe spitting out objects that have two possible features - being a raven and being black, or b) use a nonuniform distribution over an infinite set of objects and integrate (sum) to get a similar Bayesian result.
- x0x0 12y ago
  What do you mean there is no uniform distribution over any infinite set? There is the uniform distribution over [0,1] which is both infinite and not even countable. I'm confused.
  - xxxyy 12y ago
    I meant a distribution with "discrete" probability, i.e. a distribution where the probabilities of singletons are all equal and nonzero, so that a simple Bayesian argument could possibly be extended. My bad for not being precise enough. Perhaps I should have stuck to natural numbers in my previous comment, otherwise yes, you can have uniform distributions with respect to some additional structure of the probability space (like [0,1] with the Lebesgue measure you suggest).

brownbat 12y ago

I like that there are 11 distinct Bayesian solutions to the problem, at least one of them claims it is "more Bayesian" than the other Bayesian solutions. I'd like to know more about that particular solution. Unfortunately the volume containing the essay retails for around $250. I'm not sure I want to know that badly... http://www.barnesandnoble.com/w/essays-in-honor-of-carl-g-hempel-nicholas-rescher/1101630822?ean=9789027700858 http://www.barnesandnoble.com/w/essays-in-honor-of-carl-g-he...