Hempel gave an argument for a conclusion that seems absurd. I first elaborated a Bayesian argument for arriving at the opposite of the absurd conclusion, and because the conclusion (non-black non-ravens say nothing about the blackness of ravens) seems at first sight reasonable, one might think the argument reasonable (which is not reasonable, because there is nothing to stop a bad argument giving a correct conclusion).
Then I showed that combining Hempel’s argument with the grue-like concept of bnonb yielded a Hempel-style argument for non-ravens of all colours being evidence for the blackness of ravens, and further extended it to show that all properties of non-ravens are evidence for all properties of ravens.
Then I took my original argument and observed that it still works after replacing “raven” and “non-raven” by “European raven” and “non-European raven”.
At this point both arguments are producing absurd results. Hempel’s has broadened to proving that everything is evidence for everything else, and mine to proving that nothing is evidence for anything else.
I shall have to work through the arguments of Korb and Gilboa to see what they yield when applied to bnonb ravens.
Meanwhile, the unanswered question is, when can an observation of one object tell you something about another object not yet observed?
Having now properly read Korb’s paper, the basic problem he points out is that to do a Bayesian update regarding a hypothesis h in the presence of new evidence e, one must calculate the likelihood ratio P(e|h)/P(e|not-h). Not-h consists of the whole of the hypothesis space excluding h. What that hypothesis space is affects the likelihood ratio. The ratio can be made equal to anything at all, for some suitable choice of the hypothesis space, by constructions similar to those of the OP.
It makes the same negative conclusion when applied to bnonb ravens, or to European and non-European ravens.
Although this settles Hempel’s paradox, it leaves unanswered a more fundamental question: how should you update in the face of new evidence? The Bayesian answer is on the face of it simple mathematics: P(e|h)/P(e|not-h). But where does the hypothesis space that defines not-h come from?
In “small world” examples of Bayesian reasoning, the hypothesis space is a parameterised family of distributions, and the prior is a probability distribution on the parameter space. New evidence will shift that distribution. If the truth is a member of that family, evidence is likely to converge on the correct parameters.
I have never seen a convincing account of how to do “large world” Bayesian reasoning, where the hypothesis space is “all theories whatsoever, even yet-unimagined ones, describing this aspect of the world”. Solomonoff induction is the least unconvincing, by virtue only of being precisely defined and having various theorems provable about it, but one of those theorems is that it is uncomputable. Until I see someone make some sort of Solomonoff-based method work to the extent of becoming a standard part of the statistician’s toolkit, I shall continue to be sceptical of whether it has any practical numerical use. How should you navigate in a large-world hypothesis space, when you notice that P(e|h) is so absurdly low that the truth, whatever it is, must be elsewhere?
Given the existence of polar bears, arctic foxes, and snow leopards, I wondered if there might be any white-feathered ravens in the colder parts of the world. A Google search indicates that while ravens are found there, they are just as black as their temperate relatives. I guess you don’t need camouflage to sneak up on corpses. Now that looks like good evidence for all ravens being black: looking in places where it is plausible that there could be white ravens, and finding ravens, but only black ones. The not-h hypothesis space has room for large numbers of white ravens in a certain type of remote place. That part of the space came from observing polar bears and the like, and imagining a similar mechanism, whatever it might be, in ravens. Finding that even there, all observed ravens are black, removes probability mass from that part of the space.
I am saying that I am confused.
Hempel gave an argument for a conclusion that seems absurd. I first elaborated a Bayesian argument for arriving at the opposite of the absurd conclusion, and because the conclusion (non-black non-ravens say nothing about the blackness of ravens) seems at first sight reasonable, one might think the argument reasonable (which is not reasonable, because there is nothing to stop a bad argument giving a correct conclusion).
Then I showed that combining Hempel’s argument with the grue-like concept of bnonb yielded a Hempel-style argument for non-ravens of all colours being evidence for the blackness of ravens, and further extended it to show that all properties of non-ravens are evidence for all properties of ravens.
Then I took my original argument and observed that it still works after replacing “raven” and “non-raven” by “European raven” and “non-European raven”.
At this point both arguments are producing absurd results. Hempel’s has broadened to proving that everything is evidence for everything else, and mine to proving that nothing is evidence for anything else.
I shall have to work through the arguments of Korb and Gilboa to see what they yield when applied to bnonb ravens.
Meanwhile, the unanswered question is, when can an observation of one object tell you something about another object not yet observed?
Having now properly read Korb’s paper, the basic problem he points out is that to do a Bayesian update regarding a hypothesis h in the presence of new evidence e, one must calculate the likelihood ratio P(e|h)/P(e|not-h). Not-h consists of the whole of the hypothesis space excluding h. What that hypothesis space is affects the likelihood ratio. The ratio can be made equal to anything at all, for some suitable choice of the hypothesis space, by constructions similar to those of the OP.
It makes the same negative conclusion when applied to bnonb ravens, or to European and non-European ravens.
Although this settles Hempel’s paradox, it leaves unanswered a more fundamental question: how should you update in the face of new evidence? The Bayesian answer is on the face of it simple mathematics: P(e|h)/P(e|not-h). But where does the hypothesis space that defines not-h come from?
In “small world” examples of Bayesian reasoning, the hypothesis space is a parameterised family of distributions, and the prior is a probability distribution on the parameter space. New evidence will shift that distribution. If the truth is a member of that family, evidence is likely to converge on the correct parameters.
I have never seen a convincing account of how to do “large world” Bayesian reasoning, where the hypothesis space is “all theories whatsoever, even yet-unimagined ones, describing this aspect of the world”. Solomonoff induction is the least unconvincing, by virtue only of being precisely defined and having various theorems provable about it, but one of those theorems is that it is uncomputable. Until I see someone make some sort of Solomonoff-based method work to the extent of becoming a standard part of the statistician’s toolkit, I shall continue to be sceptical of whether it has any practical numerical use. How should you navigate in a large-world hypothesis space, when you notice that P(e|h) is so absurdly low that the truth, whatever it is, must be elsewhere?
Given the existence of polar bears, arctic foxes, and snow leopards, I wondered if there might be any white-feathered ravens in the colder parts of the world. A Google search indicates that while ravens are found there, they are just as black as their temperate relatives. I guess you don’t need camouflage to sneak up on corpses. Now that looks like good evidence for all ravens being black: looking in places where it is plausible that there could be white ravens, and finding ravens, but only black ones. The not-h hypothesis space has room for large numbers of white ravens in a certain type of remote place. That part of the space came from observing polar bears and the like, and imagining a similar mechanism, whatever it might be, in ravens. Finding that even there, all observed ravens are black, removes probability mass from that part of the space.