Hard questions are often hard because different ways to think about them conflict. When each way seems to have strong support, we are reluctant to choose. But if we cannot avoid the conflict, choose we must. For example, last October
Hal, I have a couple of ideas where priors come from, and I'd be interested to hear your thoughts on them.
Robin, I really like this observation that many interesting implications follow just from the idea that information is whatever allows us to exclude possibilities, and do not depend on the more controversial parts of Bayesianism. Thanks!
It is quite mysterious to me where priors come from, and seems unfortunate to have such an instability at the foundation of the Bayesian reasoning system.
It really is unfortunate -- and even more unfortunate is the fact that all reasoning systems have that instability. Even the PAC framework and SVMs are grounded in assumptions about the data-generating mechanism. I'm not (100 - ɛ)% sure of this, but my understanding is that the NFL theorems imply that learning or optimization is pretty much impossible without making some kind of structural assumption.
I personally don't worry too much about where priors come from in the general sense; as Andrew Gelman says, they come from the same place likelihoods come from. (I do carefully consider the appropriateness of the priors and likelihoods I actually put to use.)
Well that resolves the circularity -- but at the expense of being able to articulate why info theory should be favored. Which brings us back, I believe, to my original question. Well this was fun. :-)
Maybe we should take the existence of priors as a hint at our need to engage -- nay embrace -- strong intuitions. What do we really know if we don't know our priors? But really the priors should not be a mystery, they're an inherited view of the world -- inarticulate, perhaps off the mark, but nonetheless reflecting the conserved wisdom of generations and generations of people before us.
Hal, info theory alone just specifies the set of states consistent with one's info, but provides no measure on that set. It is probability theory that goes further to use a measure.
How about prior probabilities in Bayesian reasoning and information theory? They play a role analogous to the axioms in a mathematical system. They are inputs into the information theory process and seem ultimately to rely on intuition. It is quite mysterious to me where priors come from, and seems unfortunate to have such an instability at the foundation of the Bayesian reasoning system.
I'm not trying to be stubborn here. I really just don't understand how or why a theory would ever have to choose one intuition over another. Why can't a theory explain what it can explain, and leave the rest as unresolved mysteries for future research? Maybe we're not disagreeing here and I simply misunderstood what you were arguing for. But I am stubborn on the point that we should all get used to not being able to understand and explain everything around us, no matter how complete our theories.
"Choosing" among intuitions is a very slippery slope that we don't want to step onto. We're better off getting pulled back and forth around an unstable equilibrium at the top of the hill sometimes.
I think rather than waiting to know what caused our intuitions to be strong (though perhaps wrong) we should not depend on them except insofar as we have reason to believe they are correct.
I understand the temptation. (It's the same that Robin has fallen into here.) My point is that we should never choose to ignore an intuition -- especially a strong one. If our theory permits us to articulate a reason why an intuition should be ignored or is irrelevant, fine. But then we need to be hyper-sensitive to any data that later emerges in contradiction to our theory, but consistent with the intuition.
When we selectively ignore intuitions in order to advance theoretical goals, we literally isolate ourselves. Isolation is dangerous -- literally.
We should thus provisionally accept the apparent implications of standard info theory, even when they conflict with other very strong intuitions. Specifically, we should provisionally accept that:<ul><li>Math shows what axioms imply, but only unconditionally truths about non-math.<li>We have no access to moral truth beyond knowing what we want and why.<li>We have no access to our own consciousness, beyond ordinary interactions.<li>Other possible worlds are just as real as ours.<li>Analytic continuations of accepted theories should be presumed to exist.<li>We have no special access to truths about God or religion.</ul>reliability(info theory) > reliability(intuitions) > reliability(connections Robin is making between intuitions and info theory)
"Anatoly, we do not know whether there are an infinite number of pennies, but even if there are not I'd say 2+2=4 is still true about them."
It's not enough to just say - an argument is more helpful than an unsupported statement. I provided two explaining why it's incorrect to say that "the axioms of 2+2=4 (not 2+2=4 itself) apply to pennies".
But more importantly (to me personally), I'd really like to understand better what you mean by the incredibly well established information theory, separate from Bayesian probabilistic reasoning. It's a strong claim, and it'd be very useful to me if elaborated - I spent a lot of time trying to understand what information is, without much success. What is the substance of "info theory" as you refer to it, and could you provide references/links/anything to some sort of strong consensus saying that it is very strongly established? (it's fine if such discussions use completely different words to refer to the same thing, or anything like that).
"Yes, we have many specific intuitions, often very strong, supporting particular beliefs that conflict with info theory. For example, regarding consciousness most feel we know we are more than just a physical system, having also non-physical "experience." "
This seems to me to be a strawman characterization of consciousness conundrums. You don't need me to post or repost the more challenging aspects here, but it would be helpful for you to acknowledge them. I think the bottom line is there is still a big black box, and that waving it away to a debate about physical vs. non-physical systems (since whatever's in that black box will probably be redefined as part of the physical system, if it doesn't already lie in what our common 2008 understanding of what the physical system is) reduces it to a strawman that you can pit against "information theory" and equate with "God", "religion", and "moral truth" (and I suppose leprecauns and the Easter bunny).
I'd put understanding consciousness, and the question of whether the answers lie outside of the information theory framework more in an analytical category of "why do we not seem to have encountered other intelligent life" and "what came before the big bang" -- in particular, I'd put the apparent individualness, conservation, and the theatre of conscious experience as being in large parts still in a black box.
This post may be a bit messy -sorry, don't have time to edit or organize it better.
Michael Martin, you cannot rule out the possibility that your intuition(s) is/are catastrophically wrong. How do you decide between them and theory? I think rather than waiting to know what caused our intuitions to be strong (though perhaps wrong) we should not depend on them except insofar as we have reason to believe they are correct.
I read this post, and the referenced earlier post, and I wasn't able to figure out in any case what Robin means.
Can you give us a specific example of a field that claims to contradict information theory, and spell out the contradiction? Ideally, show a calculation for which info theory and the "standard practitioner of the field" get different results.
I am indeed so reluctant. If an intuition is strong, then it must be for some reason. If you don't know the reason, then you cannot rule out the possibility that your theory is catastrophically wrong.
The strength of a theory depends crucially on its ability to withstand the assault of contrary intuitions.
Hal, I have a couple of ideas where priors come from, and I'd be interested to hear your thoughts on them.
Robin, I really like this observation that many interesting implications follow just from the idea that information is whatever allows us to exclude possibilities, and do not depend on the more controversial parts of Bayesianism. Thanks!
It is quite mysterious to me where priors come from, and seems unfortunate to have such an instability at the foundation of the Bayesian reasoning system.
It really is unfortunate -- and even more unfortunate is the fact that all reasoning systems have that instability. Even the PAC framework and SVMs are grounded in assumptions about the data-generating mechanism. I'm not (100 - ɛ)% sure of this, but my understanding is that the NFL theorems imply that learning or optimization is pretty much impossible without making some kind of structural assumption.
I personally don't worry too much about where priors come from in the general sense; as Andrew Gelman says, they come from the same place likelihoods come from. (I do carefully consider the appropriateness of the priors and likelihoods I actually put to use.)
@Robin
Well that resolves the circularity -- but at the expense of being able to articulate why info theory should be favored. Which brings us back, I believe, to my original question. Well this was fun. :-)
@Hal
Maybe we should take the existence of priors as a hint at our need to engage -- nay embrace -- strong intuitions. What do we really know if we don't know our priors? But really the priors should not be a mystery, they're an inherited view of the world -- inarticulate, perhaps off the mark, but nonetheless reflecting the conserved wisdom of generations and generations of people before us.
Seed my added above.
Hal, info theory alone just specifies the set of states consistent with one's info, but provides no measure on that set. It is probability theory that goes further to use a measure.
How about prior probabilities in Bayesian reasoning and information theory? They play a role analogous to the axioms in a mathematical system. They are inputs into the information theory process and seem ultimately to rely on intuition. It is quite mysterious to me where priors come from, and seems unfortunate to have such an instability at the foundation of the Bayesian reasoning system.
Robin,
I'm not trying to be stubborn here. I really just don't understand how or why a theory would ever have to choose one intuition over another. Why can't a theory explain what it can explain, and leave the rest as unresolved mysteries for future research? Maybe we're not disagreeing here and I simply misunderstood what you were arguing for. But I am stubborn on the point that we should all get used to not being able to understand and explain everything around us, no matter how complete our theories.
"Choosing" among intuitions is a very slippery slope that we don't want to step onto. We're better off getting pulled back and forth around an unstable equilibrium at the top of the hill sometimes.
Michael, choosing info theory over a specific intuition when drawing a conclusion is not at all the same as "ignoring" that intuition.
Hopefully, a one line description pretty much has to be a "strawman."
@TGGP
I think rather than waiting to know what caused our intuitions to be strong (though perhaps wrong) we should not depend on them except insofar as we have reason to believe they are correct.
I understand the temptation. (It's the same that Robin has fallen into here.) My point is that we should never choose to ignore an intuition -- especially a strong one. If our theory permits us to articulate a reason why an intuition should be ignored or is irrelevant, fine. But then we need to be hyper-sensitive to any data that later emerges in contradiction to our theory, but consistent with the intuition.
When we selectively ignore intuitions in order to advance theoretical goals, we literally isolate ourselves. Isolation is dangerous -- literally.
Phil, that was exactly the point I was making: the problem is Robin's argument attempting to show that these things conflict with info theory.
Robin wrote:
We should thus provisionally accept the apparent implications of standard info theory, even when they conflict with other very strong intuitions. Specifically, we should provisionally accept that:<ul><li>Math shows what axioms imply, but only unconditionally truths about non-math.<li>We have no access to moral truth beyond knowing what we want and why.<li>We have no access to our own consciousness, beyond ordinary interactions.<li>Other possible worlds are just as real as ours.<li>Analytic continuations of accepted theories should be presumed to exist.<li>We have no special access to truths about God or religion.</ul>reliability(info theory) > reliability(intuitions) > reliability(connections Robin is making between intuitions and info theory)
Robin,
"Anatoly, we do not know whether there are an infinite number of pennies, but even if there are not I'd say 2+2=4 is still true about them."
It's not enough to just say - an argument is more helpful than an unsupported statement. I provided two explaining why it's incorrect to say that "the axioms of 2+2=4 (not 2+2=4 itself) apply to pennies".
But more importantly (to me personally), I'd really like to understand better what you mean by the incredibly well established information theory, separate from Bayesian probabilistic reasoning. It's a strong claim, and it'd be very useful to me if elaborated - I spent a lot of time trying to understand what information is, without much success. What is the substance of "info theory" as you refer to it, and could you provide references/links/anything to some sort of strong consensus saying that it is very strongly established? (it's fine if such discussions use completely different words to refer to the same thing, or anything like that).
"Yes, we have many specific intuitions, often very strong, supporting particular beliefs that conflict with info theory. For example, regarding consciousness most feel we know we are more than just a physical system, having also non-physical "experience." "
This seems to me to be a strawman characterization of consciousness conundrums. You don't need me to post or repost the more challenging aspects here, but it would be helpful for you to acknowledge them. I think the bottom line is there is still a big black box, and that waving it away to a debate about physical vs. non-physical systems (since whatever's in that black box will probably be redefined as part of the physical system, if it doesn't already lie in what our common 2008 understanding of what the physical system is) reduces it to a strawman that you can pit against "information theory" and equate with "God", "religion", and "moral truth" (and I suppose leprecauns and the Easter bunny).
I'd put understanding consciousness, and the question of whether the answers lie outside of the information theory framework more in an analytical category of "why do we not seem to have encountered other intelligent life" and "what came before the big bang" -- in particular, I'd put the apparent individualness, conservation, and the theatre of conscious experience as being in large parts still in a black box.
This post may be a bit messy -sorry, don't have time to edit or organize it better.
Michael Martin, you cannot rule out the possibility that your intuition(s) is/are catastrophically wrong. How do you decide between them and theory? I think rather than waiting to know what caused our intuitions to be strong (though perhaps wrong) we should not depend on them except insofar as we have reason to believe they are correct.
I read this post, and the referenced earlier post, and I wasn't able to figure out in any case what Robin means.
Can you give us a specific example of a field that claims to contradict information theory, and spell out the contradiction? Ideally, show a calculation for which info theory and the "standard practitioner of the field" get different results.
@Robin
I am indeed so reluctant. If an intuition is strong, then it must be for some reason. If you don't know the reason, then you cannot rule out the possibility that your theory is catastrophically wrong.
The strength of a theory depends crucially on its ability to withstand the assault of contrary intuitions.