As a special case of this, we can also handle noise using repeated experiments. If I roll a die, I can’t predict the outcome perfectly, so I can’t rule out influences from all the billions of variables in the universe. But if I roll a die a few thousand times, then I can approximately-perfectly predict the distribution of die-rolls (including the mean, variance, etc). So, even though I don’t know what influences any one particular die roll, I do know that nothing else in the universe is relevant to the overall distribution of repeated rolls (at least to within some small error margin).
I’m not sure I fully understand this, so I wanna try to sketch out an example to see.
Suppose you’ve got a family of unknown variables X0, X1, X2, … which each influence the observable variables Y0, Y1, Y2, …. Given some observations for some of the Yis, you can learn some summary statistics Yg that you can use to predict others Yis.
I think the counterintuitive thing about this view then, is that Yi is not independent of Xi given Yg. So what have we really learned? It doesn’t immediately tell us anything about the Xi/Yi relationship. So where’s the science?
I think my answer to this question is that while Yg doesn’t tell us anything about the Xis, it does tell us things about the Yis. (And Yg would essentially be a measure of the common causes underlying the Yis, I suppose.) Which is useful if you care about things that are downstream from the Yis. But I don’t really see what determinism buys you here.
My model of you says that you’d mention something about the KPD theorems. But I don’t know what.
Or should this more be understood in a nested sense?
That is, if you’ve got Y00, Y01, Y10, Y11, … then you can form Yg0, Yg1, …, and if Ygi is then deterministically predictable from Xi, you know you’re onto something?
I’m not sure I fully understand this, so I wanna try to sketch out an example to see.
Suppose you’ve got a family of unknown variables X0, X1, X2, … which each influence the observable variables Y0, Y1, Y2, …. Given some observations for some of the Yis, you can learn some summary statistics Yg that you can use to predict others Yis.
I think the counterintuitive thing about this view then, is that Yi is not independent of Xi given Yg. So what have we really learned? It doesn’t immediately tell us anything about the Xi/Yi relationship. So where’s the science?
I think my answer to this question is that while Yg doesn’t tell us anything about the Xis, it does tell us things about the Yis. (And Yg would essentially be a measure of the common causes underlying the Yis, I suppose.) Which is useful if you care about things that are downstream from the Yis. But I don’t really see what determinism buys you here.
My model of you says that you’d mention something about the KPD theorems. But I don’t know what.
Or should this more be understood in a nested sense?
That is, if you’ve got Y00, Y01, Y10, Y11, … then you can form Yg0, Yg1, …, and if Ygi is then deterministically predictable from Xi, you know you’re onto something?