You might think this is "unfair": some women—about 7 per 1000—are 5′11″, and we don't want to say they're somehow _less female_ on that account, so why can't I keep my height? The problem is that if we refuse to adjust for every trait for which the female and male distributions overlap (on the grounds that _some_ women have the same trait value as my male self), we don't end up with a result from the female distribution.
-The typical point in a high-dimensional distribution is _not_ typical along each dimension individually. [In 100 flips of a biased coin](http://zackmdavis.net/blog/2019/05/the-typical-set/) that lands Heads 0.6 of the time, the _single_ most likely sequence is 100 Heads, but there's only one of those and you're _vanishingly_ unlikely to actually see it. The [sequences you'll actually observe will have close to 60 Heads](https://en.wikipedia.org/wiki/Asymptotic_equipartition_property). Each such sequence is individually less probable than the all-Heads sequence, but there are vastly more of them. Similarly, [most of the probability-mass of a high-dimensional multivariate normal distribution is concentrated in a thin "shell" some distance away from the mode](https://www.johndcook.com/blog/2011/09/01/multivariate-normal-shell/), for the same reason. (The _same_ reason: the binomial distribution converges to the normal in the limit of large _n_.)
+<a id="typical-point"></a>The typical point in a high-dimensional distribution is _not_ typical along each dimension individually. [In 100 flips of a biased coin](http://zackmdavis.net/blog/2019/05/the-typical-set/) that lands Heads 0.6 of the time, the _single_ most likely sequence is 100 Heads, but there's only one of those and you're _vanishingly_ unlikely to actually see it. The [sequences you'll actually observe will have close to 60 Heads](https://en.wikipedia.org/wiki/Asymptotic_equipartition_property). Each such sequence is individually less probable than the all-Heads sequence, but there are vastly more of them. Similarly, [most of the probability-mass of a high-dimensional multivariate normal distribution is concentrated in a thin "shell" some distance away from the mode](https://www.johndcook.com/blog/2011/09/01/multivariate-normal-shell/), for the same reason. (The _same_ reason: the binomial distribution converges to the normal in the limit of large _n_.)
Statistical sex differences are like flipping two different collections of coins with different biases, where the coins represent various traits. Learning the outcome of any individual flip, doesn't tell you which set that coin came from, but [if we look at the aggregation of many flips, we can get _godlike_ confidence](https://www.lesswrong.com/posts/cu7YY7WdgJBs3DpmJ/the-univariate-fallacy-1) as to which collection we're looking at.
projects / Ultimately_Untrue_Thought.git / blobdiff