There’s a thing I keep thinking of when it comes to natural latents. In social science one often speaks of statistical sex differences, e.g. women tend to be nicer than men. But these sex differences aren’t deterministic, so one can’t derive them from simply meeting 1 woman and 1 man. It feels like this is in tension with the insensitivity condition, like that this only permits universal generalizations (I guess stuff like anatomy?). But also I haven’t really practiced natural latents at all so I sort of expect to be using them in a suboptimal way. Maybe you have a better idea?
Like I mean obviously one can deterministically recover it when one considers large samples of people, so it seems like maybe the “statistical man” and “statistical woman” concepts are different from the “universal man” and “universal woman” concepts. Not entirely sure this is a good approach though.
It feels like this is in tension with the insensitivity condition, like that this only permits universal generalizations (I guess stuff like anatomy?)
Quite the opposite! In that example, what the insensitivity condition would say is: if I get a big sample of people (roughly 50⁄50 male/female), and quantify the average niceness of men and women in that sample, then I expect to get roughly the same numbers if I drop any one person (either man or woman) from the sample. It’s the statistical average which has to be insensitive; any one “sample” can vary a lot.
That said, it does need to be more like a universal generalization if we impose a stronger invariance condition. The strongest invariance condition would say that we can recover the latent from any one “sample”, which would be the sort of “universal generalization” you’re imagining. Mathematically, the main thing that would give us is much stronger approximations, i.e. smaller ϵ’s.
There’s a thing I keep thinking of when it comes to natural latents. In social science one often speaks of statistical sex differences, e.g. women tend to be nicer than men. But these sex differences aren’t deterministic, so one can’t derive them from simply meeting 1 woman and 1 man. It feels like this is in tension with the insensitivity condition, like that this only permits universal generalizations (I guess stuff like anatomy?). But also I haven’t really practiced natural latents at all so I sort of expect to be using them in a suboptimal way. Maybe you have a better idea?
Like I mean obviously one can deterministically recover it when one considers large samples of people, so it seems like maybe the “statistical man” and “statistical woman” concepts are different from the “universal man” and “universal woman” concepts. Not entirely sure this is a good approach though.
Quite the opposite! In that example, what the insensitivity condition would say is: if I get a big sample of people (roughly 50⁄50 male/female), and quantify the average niceness of men and women in that sample, then I expect to get roughly the same numbers if I drop any one person (either man or woman) from the sample. It’s the statistical average which has to be insensitive; any one “sample” can vary a lot.
That said, it does need to be more like a universal generalization if we impose a stronger invariance condition. The strongest invariance condition would say that we can recover the latent from any one “sample”, which would be the sort of “universal generalization” you’re imagining. Mathematically, the main thing that would give us is much stronger approximations, i.e. smaller ϵ’s.
Oops, my bad for focusing on the simplified version and then extrapolating incorrectly.