Comments on: Probabilistic Inference Barrier http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/ A general distrust of strong metaphysical claims in mathematics and philosophy. Tue, 14 Jul 2009 21:45:18 +0000 http://wordpress.com/ hourly 1 By: Kenny http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-2194 Kenny Fri, 24 Aug 2007 07:46:54 +0000 http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-2194 I should also compare this to a point that Malcolm Forster attributes to Karl Popper in his note on David Deutsch's paper on decision theoretic probabilities in quantum mechanics. He suggests that Popper at some point said "no probabilities in, no probabilities out" in The Logic of Scientific Discovery. This result is some sort of converse to that. I should also compare this to a point that Malcolm Forster attributes to Karl Popper in his note on David Deutsch’s paper on decision theoretic probabilities in quantum mechanics. He suggests that Popper at some point said “no probabilities in, no probabilities out” in The Logic of Scientific Discovery. This result is some sort of converse to that.

]]> By: Mike Titelbaum http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-352 Mike Titelbaum Wed, 24 Jan 2007 09:28:43 +0000 http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-352 The reason I asked is that I'm not sure how your construction works. You claim that O is false in P'. Clearly if P' designates a state as actual and that state assigns falsehood to O, O is false on P'. But in your response you wrote, "whether X is actual or not is in some sense irrelevant to its probability - if a state couldn't be the actual one, then it shouldn't be in your model at all." That makes it sound like what's important here is that P' is defined over a state space one of whose states assigns falsehood to O, even though P' assigns a probability of zero to that state. If that's what the argument turns on (instead of the fact that P' designates X as actual), I'm wondering why those facts make O *false* on P'. The reason I asked is that I’m not sure how your construction works. You claim that O is false in P’. Clearly if P’ designates a state as actual and that state assigns falsehood to O, O is false on P’. But in your response you wrote,

“whether X is actual or not is in some sense irrelevant to its probability – if a state couldn’t be the actual one, then it shouldn’t be in your model at all.”

That makes it sound like what’s important here is that P’ is defined over a state space one of whose states assigns falsehood to O, even though P’ assigns a probability of zero to that state. If that’s what the argument turns on (instead of the fact that P’ designates X as actual), I’m wondering why those facts make O *false* on P’.

]]> By: Kenny http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-351 Kenny Mon, 22 Jan 2007 23:14:29 +0000 http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-351 Also, I think I should specify that both sorts of arguments only work if the probability statements are unconditional, rather than conditional. Conditional probability values can change arbitrarily even with the introduction only of very improbable states. And they need even further specification if states of probability 0 exist. Also, I think I should specify that both sorts of arguments only work if the probability statements are unconditional, rather than conditional. Conditional probability values can change arbitrarily even with the introduction only of very improbable states. And they need even further specification if states of probability 0 exist.

]]> By: Kenny http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-350 Kenny Mon, 22 Jan 2007 23:11:59 +0000 http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-350 I think that infinite cases are helpful motivation for possible states of probability 0, but that once we recognize this conceptual possibility, the infinity isn't necessary. Of course, I also think that there <i>are</i> infinitely many states in the space that most of us work with, because I think of states as giving maximally specific descriptions of how things might be. But at any rate, whether X is actual or not is in some sense irrelevant to its probability - if a state couldn't be the actual one, then it shouldn't be in your model at all. And even if probability 0 states aren't allowed, the implication barrier between probabilistic strict inequalities and statements of fact will hold, because the new state can be introduced with a probability even smaller than any difference relevant to the inequalities. Unless, by analogy to barring probability 0 unless there are infinitely many states, one bars really small probabilities unless there are enough (but still only finitely many) states. I think that infinite cases are helpful motivation for possible states of probability 0, but that once we recognize this conceptual possibility, the infinity isn’t necessary. Of course, I also think that there are infinitely many states in the space that most of us work with, because I think of states as giving maximally specific descriptions of how things might be.

But at any rate, whether X is actual or not is in some sense irrelevant to its probability – if a state couldn’t be the actual one, then it shouldn’t be in your model at all.

And even if probability 0 states aren’t allowed, the implication barrier between probabilistic strict inequalities and statements of fact will hold, because the new state can be introduced with a probability even smaller than any difference relevant to the inequalities.

Unless, by analogy to barring probability 0 unless there are infinitely many states, one bars really small probabilities unless there are enough (but still only finitely many) states.

]]> By: Mike Titelbaum http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-349 Mike Titelbaum Mon, 22 Jan 2007 11:04:47 +0000 http://antimeta.wordpress.com/2007/01/21/probabilistic-inference-barrier/#comment-349 Hey Kenny, Your argument requires constructing a space in which P'(X)=0 despite the fact that X is the "actual" state. This is plausible in situations with uncountably many mutually incompatible events, but do you think it's acceptable in situations where everything is finite? If not, does your inference (really implication) barrier result apply to such finite situations? Hey Kenny,
Your argument requires constructing a space in which P’(X)=0 despite the fact that X is the “actual” state. This is plausible in situations with uncountably many mutually incompatible events, but do you think it’s acceptable in situations where everything is finite? If not, does your inference (really implication) barrier result apply to such finite situations?

]]>