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Hmm, I think when the author gets to imagining an "infinite dataframe" to deal with conditioning with continuous variables, it's kind of revealed that this isn'
by abeppu 3y ago
Hmm, I think when the author gets to imagining an "infinite dataframe" to deal with conditioning with continuous variables, it's kind of revealed that this isn't the best way to build intuition. Worse, sometimes we talk about conditioning on _unobserved_ variables which wouldn't be in your dataframe. This happens when we describe a model with latent/hidden variables that impact the stuff we do observe.
I think a conventional way of describing this, which I think is in intro textbooks, is better -- we thinking of cutting slice through a multivariate distribution, and then normalizing. If you have some distribution P(X, Y) where X and Y are both real, then its density is like some landscape (i.e. each x,y is associated with some height). P(X | Y=y) is like a slice through that landscape at Y=y, which leaves a 1-D plot of the density along X. If we re-scale it so that the area under it is 1, then it's a well-formed PDF.
- refulgentis 3y agoOh, I see!
- ShamelessC 3y ago> we thinking of cutting slice through a multivariate distribution, and then normalizing. If you have some distribution P(X, Y) where X and Y are both real, then its density is like some landscape (i.e. each x,y is associated with some height). P(X | Y=y) is like a slice through that landscape at Y=y, which leaves a 1-D plot of the density along X. If we re-scale it so that the area under it is 1, then it's a well-formed PDF. How intuitive! /s
- markisus 3y agoThis is a case where a picture is worth a thousand words. The slicing intuition is quite useful and I imagine quite common for those who have taken calculus. It is the very same mental picture as you would use for computing area using two nested integrals. This video [1] has some nice graphics along these lines. [1] https://www.youtube.com/watch?v=gifQWtTWqEY https://www.youtube.com/watch?v=gifQWtTWqEY