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Slider comments on Open Thread, Jun. 29 - Jul. 5, 2015 - Less Wrong Discussion
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The surprise comes only to those that try to overload probability with roles it should not have. For example "A implies B" does imply P(B|A)=1 but P(B|A)=1 doesn't imply "A implies B". While it is common that if we know systematically that a certain probability is high it is a promising line of argument for causations and implications it doesn't always carry through (meta-example I am in essence arguing that while an okayish rule of thump, it is actually improper to infer causations from probabilities. I am doing this by pointing out that P(causation|probability) ~ 1 and P(causation|probability)<1 which alone is only suggestive that it is so (or the actual steps are implied)).
Can you give a counterexample?
A: "X is a random number, uniformly distributed on the interval from 0 to 1."
B: "X is irrational."
Then Pr(B|A) = 1 because almost all numbers are irrational (more formally: because the rationals have measure 0), but A doesn't imply B because X could be rational.
This could be made to not be a counterexample by using a theory of probability that uses surreals.
That is Pr(irrational|random form 0 to 1) being 1 is of the "almost always" kind, which can be separated form the kind of 1 that is of the "always" kind.
for ω that is larger than any surreal that has a real-counter part, there is a ɛ=1/ω.
Taking only finite samples out of a infinite group makes for a probability that is smaller than any real probability that could well be represented by real/natural multiples of ɛ.
Similarly taking only countably infinite samples from a group of uncountably many samples would result in a probability larger than 0 but smaller than any real value.
Thus we could have P(irrational|between 0 and 1)=1-xɛ and P(rational|between 0 and 1)=xɛ that would sum to exactly 1 and yet P(Z|0-1)=xɛ>0 ie a positive probability.
Similarly the probability of a dart landing exactly on a line in a dart board is "almost never" ie 0 yet that place is as probable as any other location on the dart board. It would be possible to find a dart exactly on the line, you would not just expect to encounter it in a finite number of throws.
However there are counterexamples where all As are indeed Bs but no implication is possible.
Surreal numbers do not yet have a good theory of integration. This makes surreal probability theory problematic.
Coextensive properties that are not the same property. There are some biological facts like these. Probably not remembering correctly but for example B="animal has heart" A="animal is mammal" it can easily be that all mammals in fact have hearts but you couldn't still say that it would be impossible for a mammal to be heartless (and for example have a blood circulation system that is evenly distributed all over the veins (which they kinda partially do but be totally reliant on those kind of mechanism)). The deduction of "It is a mammal, it must have a heart" is false for plenty of reasonable senses of "must". It is true for the probabilistic sense of must but implication has more senses than the probabilistic one.
If it's a given that all mammals have hearts, then being a mammal implies it has a heart. If it's not known that all mammals have hearts, then P(B|A) < 1.