As someone who finds most mathematics broadly interesting, but
doesn’t know enough about statistics to comprehend much past the
introductory paragraph of the relevant Wikipedia article, would a kind
fellow sneerer please explain? Is this wrong, technically true but
irrelevant, or something else?
Central Limit Theorem comes in a few flavours - some very obscure versions in very specific areas sometimes don’t require independence, but for the main version that everyone refers to and uses 99.99% of the time independence is absolutely a critical assumption so he’s dead wrong - like not even a technical mixup it’s just fundamental to the idea of what the CLT is saying
So you are saying Hanson is right in 0.001% of the cases? So technically he isn't wrong. [Checkmate sneertrooper!](https://www.youtube.com/watch?v=hou0lU8WMgo)
(But thanks for explaining)
To add from the common-sense side of things: Hanson is using the theorem in a word game to establish a particular point of view he already holds. He isn’t actually using the theorem as an example of a statistical proof of his opinion.
Oh yeah, that much I figured. If he tweeted the Pythagorean theorem, with no additional comment, I’d wonder what it was about the law of cosines that was inconvenient for his narrative.
Technically true but irrelevant. There are [relaxations of the CLT that don't require full independence](https://en.wikipedia.org/wiki/Central_limit_theorem#Dependent_processes), but if you remove the assumption completely then it's trivially false (e.g., sample a random variable X_1 from the uniform distribution over [0, 1], then take X_2 = X_1, X_3 = X_1, X_4 = X_1, and so on up to X_n --- these are all uniformly distributed over [0, 1] and their sum is uniformly distributed over [0, n]).
I haven't been following the blow-by-blow of this debate but my guess is that Hanson's first impulse was "assume independence, what could go wrong", then when people called him on it, he checked Wikipedia, saw the existence of generalizations, then figured he could just slot one of those into his argument and keep going.
Yeah my understanding is that it’s an ongoing area of research what niches you can drop the independence condition in, but the more common usage in terms of assuming sums/averages tend towards normal distributions does absolutely require independence right?
Honestly, I know less about this than I should, or would like to. (I've read more than one paper where the author cites a [martingale convergence theorem](https://en.wikipedia.org/wiki/Doob%27s_martingale_convergence_theorems), handwaves furiously, then alleges a bizarre conclusion about epistemology....)
You can fully remove the assumption (and the need for a limit) if you know your initial random variables are normally distributed.
However, at this point it's not really the central limit theorem and more the "adding normal distributions together gives another normal distribution" theorem.
Neat. I'd not seen that before. Thanks.
So, you can take a copy of a normal distribution sign flip an restricted interval about the mean; and leave the tails untouched. The resulting variable is still normal but when you add it to the original variable all samples in the restricted interval go to zero, and the tails just double.
https://en.m.wikipedia.org/wiki/Multivariate_normal_distribution#Two_normally_distributed_random_variables_need_not_be_jointly_bivariate_normal
>these are all uniformly distributed over [0, 1] and their sum is uniformly distributed over [0, n]
Shouldn´t they again be uniformly distributed over [0,1]? I don´t see where [0,n] gets in here at all.
As far back as Rousseau - or even much further back to Socrates - you can find arguments that democracy and full participation in democracy is a good in itself: the demos is just insofar as it is able to exist at all, rather than simply because a utilitarian ethic justifies it.
This is, at least, what I believe and I think there are strong reasons to believe it.
Moreover, particular instantiations of the “low information voter” concept have tended to come with horrific prejudices: black people; poor people; you get what I mean. And those people have interests which regardless of the overall utility function of a society need to be represented within the demos, not least because when they aren’t Bad Shit tends to happen. The argument that “low information voters” (whoever they are) should abstain is a first-shot kind of argument that never gets to the next step of examining the historical context in which democracy itself emerged in human societies.
I mean the most practical reasons are just that there’s not a good objective criterion for what counts as being sufficiently informed for an election, nor is there any good way for people to reliably self-assess
GMU tenure, the gift that keeps on giving.
As someone who finds most mathematics broadly interesting, but doesn’t know enough about statistics to comprehend much past the introductory paragraph of the relevant Wikipedia article, would a kind fellow sneerer please explain? Is this wrong, technically true but irrelevant, or something else?
Found this in one of the comment threads on one of his weird diatribes about how low-information voters should abstain
What the hell?