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RichardKennaway comments on Should correlation coefficients be expressed as angles? - Less Wrong Discussion

43 Post author: Sniffnoy 28 November 2012 12:05AM

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Comment author: RichardKennaway 28 November 2012 08:44:05AM 3 points [-]

In some cases, a (positive) correlation coefficient can be thought of as a probability, which is another point in its favor.

I haven't come across that. Could you amplify it?

Comment author: Kindly 28 November 2012 01:04:00PM *  2 points [-]

Edit: I started out giving a somewhat confusing example so I'm just going to edit this comment to say something less confusing.

Let X be any random variable, and define Y as follows: with probability R, Y=X, and otherwise Y is independently drawn from the same distribution as X. Then X and Y are identically distributed and have correlation R.

Comment author: Douglas_Knight 28 November 2012 10:25:04PM 0 points [-]

What definition of correlation are you using?

Comment author: Kindly 28 November 2012 10:39:16PM *  0 points [-]

This one. As far as I know, it's the only kind of correlation that the cosine interpretation is valid for.

If you want to verify my claim, it will help to assume that your distribution has mean 0 and variance 1, in which case the correlation between X and Y is just E[XY]. But correlation is invariant under shifting and scaling, so this is fully general.

Edit: I suppose I was imprecise when referring to the correlation between bit strings; what I mean there is simply the correlation between any pair of corresponding bits. This sort of confusion appears to be standard.

Comment author: Douglas_Knight 28 November 2012 11:31:32PM 0 points [-]

Yes, I was wondering what you meant by the correlation of a string. Also, the definition doesn't apply to binary variables unless you somehow interpret bits as numbers, but as you say, it doesn't matter how you do so. There's a reason I asked what definition you were using, not the original post.

This sort of confusion appears to be standard.

Which confusion? The confusion between a random variable and sequence of iid draws from it? And what do you mean by standard?

Comment author: Kindly 28 November 2012 11:55:20PM *  0 points [-]

By "this sort of confusion" I meant the ambiguity between the correlation of two bit strings and the correlations between the individual bits. By "standard" I mean that I didn't make this up; I've seen other people do this.

Anyway, perhaps it's better to focus on the more general (and less notation-abusing) example I gave.