Well, if it matters at all - I didn’t inhale.
Funny, I was nodding off half way through the OP.
In fact, if we hold the same posterior probabilities after having seen finite data, then we must have held the same priors to begin with.
It isn’t clear to me what role “data” is playing in this process of updating. For example, assume P(A) = .2, P(B) = .5. In the face of data, A, since P(A|A) = 1, it seems natural that the revision would revise P(A) to 1. In that case, P(A|B) = .5. And, for any non-zero probability of A, the revision process would end up with P(A) = 1, P(B) = .5.
That’s not Bayesian reasoning. The Bayesian approach to this problem would be to say–as an example, mind you–that P(A) and P(B) are marginally uniform on [0, 1] with the constraint that P(A) < P(B). Once you see the data, you can make additional inferences on P(A) and P(B).
I’m not clear on what counts as data. Your example seems to be a case of learning more about (how probabilities) relate to each other
Is the data just the prior probabilities, run through an updating of relative probabilities? Is it a change in the prior probabilities? Am I missing the point entirely?
For some reason I though there was more to “Bayesian updating” than just posterior probability, but now see that Half Man Half Wit meant just that. In that case, given we disagree about prior probabilities, if we have some A where P(A) = 1, then P(B|A) = P(B). Given that assumption it certainly follows that any different prior probability distribution will be different from a posterior probability distribution. I do not know if prior probabilities are just special cases of posterior probabilities in general.
“Extraordinary claims require extraordinary evidence” is an oft-recycled phrase, particularly within the skeptical community. It gets recycled a lot just because it’s attributed to Carl Sagan, which is odd because this comes close to the kind of ‘argument from authority’ that skeptics and critical thinkers are often so keen to despise.
It’s not a very wise phrase.
Evidence does not come in two flavours: ordinary and extraordinary. If anyone thinks otherwise, I’d like to see which litmus test you apply to distinguish one from the other. There is just evidence that supports a claim, or evidence that contradicts it. That’s all. So the term ‘extraordinary evidence’ is nonsense.
Also, the assertion that “claims” require evidence is open to question. I wasn’t aware that “claims” require anything at all. People investigating a claim might want to see some evidence, but if that’s what we mean then that what we ought to say. And if you think I’m nit-picking, well, that’s what critical thinkers and skeptics often assert is exactly the right thing to do: pay attention to small details in case they are important.
What Sagan meant was, “The more extraordinary a claim is, the more appropriate it is to reserve judgment until we’ve seen good supporting evidence”. Even knowing that this is the case, skeptics keep recycling the quoted version, even though it’s nonsense, because they’ve seen it attributed to Carl Sagan and therefore feel it must contain great wisdom that mustn’t be challenged or corrected (much like religious followers won’t challenge any piece of scripture, even when it’s completely bonkers).
Well, the model is somewhat simplified, of course – in particular, it assumes the interpretation of the data to be trivial, as in the cookie example: there’s no matter of interpretation whether or not a cookie is a chocolate chip cookie, it either is or it’s not. Real world cases are of course typically more complex, and one can quibble how well data is in accordance with the hypothesis, and so on, but for the conceptual points I wanted to make, that would just have been excess baggage.
But if you grant the possibility of unambiguous data interpretation, then the method I’ve given to assess the validity of hypothesis is the rational one; if you find a chocolate chip cookie in the second example, and assign to the hypothesis of possessing jar A a probability different from 36%, then you’re not acting rationally.
Well, there’s some kind of infinitary flavor to it, of course, but Zeno’s problem was that his reasoning was simply wrong – one can sum up infinitely many parts and obtain a finite whole. But one can’t reach identical posteriors with finite data.
Of course, but it’s also a widespread belief that there exists some kind of ‘killer argument’ or demonstration that must suffice for every rational agent, that it is possible to make a case so well everybody ought to be convinced; but that’s not true in general (though it will be true in the vast majority of cases). It’s perhaps not something of great practical importance, but I think a conceptual point that’s too often neglected.
Well then that may be the problem!
Yes, exactly, and moreover, for any finite amount of data, there exist priors that make our disagreement arbitrarily large.
Well, ‘A’ is ‘the hypothesis is true’, so if you receive A as data, then obviously you must judge the hypothesis true; but typically, there’s no data of that kind available, but rather observations that are in variable amounts of agreement with the hypothesis (such as drawing a chocolate chip cookie: it’s possible under both hypotheses, but it’s in better agreement with that hypothesis in which the probability of drawing a chocolate chip cookie is higher); the Bayesian procedure allows you to quantify your belief in, or degree of acceptance of, the hypothesis given such data.
As I’ve said, I didn’t mean ‘extraordinary’ to mean ‘different from ordinary evidence in kind’, just that you need to present an extraordinary amount of evidence to support claims in stark contrast with prior knowledge, or justified belief. Perhaps I should have stuck with Laplace’s version: “The weight of evidence for an extraordinary claim must be proportioned to its strangeness.” But then it lacks the brevity and clarity that makes the attributed-to-Sagan version so memorable.
As an example, in particle physics, a statistical significance of five standard deviations is usually considered enough to claim a discovery; the recent findings of apparent superluminal neutrino motion exceeded that threshold, yet nobody, not even the scientists responsible for the experiment, claimed it as any sort of ‘discovery’ – because given the context, i.e. the well supported theories underlying the fundamental nature of the speed-of-light limit, you need more to overthrow the standard view.
To make a claim typically entails asserting the truth of that claim, so I think it’s valid to request evidence to back up that claim, for instance.
You may be in near agreement that we certainly went to the moon, but without knowing the qualifications of the two people assessing the information(or even knowing what information they possess in the first place), it doesn’t help us know the facts of the matter one bit. A consensus of people’s opinions doesn’t matter-a consensus of facts do.
Thanks for the extremely lucid and informative explanation of prior and posterior probabilities.
Regarding this last point, I don’t think it follows–for not only do no two people have the same priors, they also don’t have the same data (because everyone’s experience is their own, and is strictly speaking a different datum than that had by anyone else) and because of this, it turns out to be possible for two people to be in complete agreement. (Their priors are different, their data are different, but the posteriors can, in this case, turn out the same.)
How probable is this total agreement, though? I don’t dare speculate.
I have always been skeptical about the “extraordinary claims/extraordinary evidence” thing but mostly because I have not found people’s conception of “extraordinary” to be very well clarified (or well thought out even).
But in light of your OP I think I’m prepared to accept at least a version of it:
“A claim with a very low prior probability shouldn’t be accepted unless supported by data which also has a very low prior probability.”
I think that’s equivalent to the extraordinary claims maxim. And right now it appears to me to be practically tautological.
I’ve read that the ability to reason did not develop in humans to assist us in understanding complex truths, but as a way for an individual to convince others to do his bidding or otherwise see his point of view. So from that perspective, debate is nothing more than a way to convince lesser ape-brains to let a superior ape-brain have his way. So if you buy that, then the best way to deal with an intransigent debater is to expose the true motivation behind the position he’s taken.
I would think that a debate about the existence of faster than light neutrinos would proceed much more reasonably than a debate on global warming. It would take a much greater level of scientific expertise in order to stake out a self-serving claim regarding neutrinos.
Ah yes, you’re very right there! Didn’t think in that direction at all, thanks for pointing it out. For any situation of significant complexity, it’s of negligible probability for this to happen (thank god for fudge words! :p), but since I’m the one harping on about things being possible in principle here, the point is taken. And if we relax ‘agreement’ to a ‘close enough for government work’ definition, this is probably the typical situation.
But is that good grounds for agreement? It seems to invite Gettier-type problems: coming to the right conclusion by the wrong path. (I have in mind something similar to a situation where, in a cafe, you hear your cell phone ringtone, pick up, and are, in fact, being called; but unbeknownst to you, your phone had been set to mute, so you didn’t actually hear it ring, and just coincidentally, a person sitting near you was being called, as well, and had his phone set to the same tone. I, on the other hand, sitting somewhere near by, could have seen your phone light up, and have concluded that you are being called – we agree on the fact that you are being called, but your way to reach that conclusion, and hence, our way to the agreement, seems a bit funny, at least, if not outright suspect.)
Right, I was just thinking about this. Couple of things to say, very speculative:
For one thing, it could be that we’re “designed” (in a broad evolutionary sense) such that these coincidences of data and priors tend to work out for us, collectively, in the long run. If that’s plausible, then it gives us a basis to trust the “coincidence of it all” to some extent.
For another thing, (completely separate from the prior speculation,) it could be that this problem simply serves to highlight the collective nature of scientific endeavors. You and I, as individuals, are not actually rationally justified in taking our substantial agreement to be evidence for the truth about what we agree about. Nevertheless, the more-or-less formal practice of actually doing measurements, calculating probabilities, etc., issues forth in a collective endeavor that is truth-oriented as a whole, over time. I think I had something a lot more eloquent and clear to say about that but between the time I started writing and the time I’m writing this I had a student come in for help and everything in my head kind of went away…
More specifically, we only accept a claim after seeing the evidence if the posterior probability that it’s true is greater than the posterior probability that it’s false. Therefore, if your prior probability on the claim is low, you’re going to need strong evidence to accept it.
OPs of extraordinary length require extraordinary quality to be worth reading. 
I’m not sure what you mean by a claim within the consensus. If it something known, it might be a good lab for science class, but someone trying to publish a paper about it won’t be too successful. If it is an extension of knowledge that falls within the consensus, it can get published but for the most part few will reference it and fewer will try to reproduce it.
Yes, all evidence is ordinary in a sense, but I interpret extraordinary evidence in this sense is evidence that has been more carefully vetted than usual and which has been reproduced. The researchers in the particle moving faster than c case are doing their best to ensure that the evidence is extraordinary by looking for possible sources of error and by reproducing the experiment. Cold fusion is a good case of an extraordinary claim made without extraordinary evidence, in contrast.
I’d say extraordinary claims are those where the truth or falsity of the claim is significant. There are tons of incorrect results out there, which mostly no one cares about. Those are the ordinary claims.
My new crusade is against the use of probabilities for cases like this. No slam on you - it is prevalent in many ares. The way to assess beliefs is Dempster-Shafer Theory.