I implicitly meant a continuous distribution. Clarified that in the post now.
Concretely, if you’re measuring the length of something with a ruler, you probably just round to the nearest 1/16th of an inch.
As someone who lives in the dangerous and uncharted part of the world called “outside the US’, I prefer centimeters. ;)
Her confusion.
When I read thakll’s post, I thought they indeed meant the mathematical definition of “almost surely”. The domain of an event with probability zero is indeed “almost nowhere” in the rigorous sense, since it is a measure-zero set.
Now, a hopefully intuitive explanation of independent events.
By definition, A is independent from B if P(A|B) = P(A), or equivalently P(AB) = P(A)P(B). What does it mean in terms of measures?
It is easy to prove that if A is independent from B, then A is also independent from ~B: P(A|~B) = P(A ~B) / P(~B) = (P(A) - P(AB)) / (1 - P(B)) = (P(A) - P(A)P(B)) / (1 - P(B)) = P(A).
Therefore, A is independent from B iff P(A) = P(AB) / P(B) = P(A ~B) / P(~B), which implies that P(AB) / P(A ~B) = P(B) / P(~B).
Geometrically, it means that A intersects B and ~B with subsets of measures proportionate to the measures of B and ~B. So if P(B) = 1⁄4, then 1⁄4 of A lies in B, and the remaining 3⁄4 in ~B. And if B and ~B are equally likely, then A lies in equal shares of both.
And from an information-theoretic perspective, this geometric interpetation means that knowing whether B or ~B happened gives us no information about the relative likelihood of A, since it will be equally likely to occur in the renormalized outcome space either way.
...Oops, yes, said that without thinking. But this
Basically, P(A|B) = 0 when A and B are disjoint, and P(A|C)/P(B|C) = P(A)/P(B) when A and B are subsets of C?
is correct.
and P(A|C)/P(B|C) = P(A)/P(B) when A and B are subsets of C?
When A is a subset of C, P(A|C) = P(A).
I agree with the OP: simply defining a probability concept doesn’t by itself map it to our intuitions about it. For example, if we defined P(A|B) = P(AB) / 2P(B), it wouldn’t correspond to our intuitions, and here’s why.
Intuitively, P(A|B) is the probability of A happening if we know that B already happened. In other words, the entirety of the elementary outcome space we’re taking into consideration now are those that correspond to B. Of those remaining elementary outcomes, the only ones that can lead to A are those that lie in AB. Their measure in absolute terms is equal to P(AB); however, their measure in relation to the elementary outcomes in B is equal to P(AB)/P(B).
Thus, P(A|B) is P(A) as it would be if the only elementary outcomes in existence were those yielding B. P(B) here is a normalizing coefficient: if we were evaluating the conditional probability of A in relation to a set of exhaustive and mutually exclusive experimental outcomes, as it is done in Bayesian reasoning, dividing by P(B) means renormalizing the elementary outcome space after B is fixed.
I’m actually staggered by the amount of so-called “dating advice” on LW in the first place.
This whole post is disturbing to me for several reasons, the least of which is the analysis of relationships—which, to me, have always been emotional, passionate moments—by talking like a machine about optimal universes and utilons. Even Spock wasn’t that cold. This is simply not what rationality means to me.
The most disturbing reason is seeing this post commit such fallacies of sexism as stereotyping and othering. Attempts to pigeonhole the mind-boggling diversity of romantic relationships into premade cookie-cutter recipes are at best doomed to failure and at worst show, if not something so drastic as objectification, at least a failure of empathy, since to the author it only extends to the “us”-group and not to the “them”-group.
May defame my image with women (who would date an immortalist after all …..)
...
I have better things to do than looking at karma points and seeing who’s considered privileged on this site and who isn’t.
Such as writing meaningful posts here. As opposed to roleplaying distracting from real content.
This, if anything, cements my belief that LW is not the right place for me.
Speak for yourself.
I think you’re taking this roleplaying thing too far.
How exactly is it wrong?
Except the agenda of the fraction I’m speaking about is not “technology will not destroy the world”. It’s “friendly AI and uploads will lead us to a bright perfect techno-utopia”. And as I said multiple times before, I don’t buy it.
How to solve the national debt deadlock
In GetDefaultCountry()?
Wouldn’t dropping the rationality tagline instead convince people even more thoroughly that it’s not actually about rationality, but rather something else?
That being said, I agree with those concerns. LW doesn’t have an agenda per se (beyond being sponsored by the SIAI), but the LW majority clearly does. While harsh, “a bunch of fringe technophiles” accurately describes a significant, and vocal, fraction of people here.
Sorry, my fault for misspelling it.
Depends. In casual use, typically centimeters. But yes, as muflax said, metric rulers have individual millimeters marked, and typically they mark half-centimeters with slightly longer bars.