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Cyan comments on The usefulness of correlations - Less Wrong
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Huh? Everything in the post still follows if you impose the condition that X, Y, and Z are all zero mean. Then b is always zero for both the (X,Y) relationship and the (X,Z) relationship. Also, "Z lies close to X" means Z = X + e, where e is a zero-mean random variable with a standard deviation a fair bit smaller than the standard deviation of X. What the heck is the lower-case "x" in your equation?
My 'mx' is the same as your 'e'.
Okay, it is also true that correlation does not account for scale. Y's correlation with X does not discern between Y = X and Y = 9X.
I still don't understand how it can be a condemnation of probability and statistics. Both the approaches use probability and statistics. The opening sentence implies that what follows compares probability and statistics to some other approach, but there isn't any other approach following it. Just correlation vs. confidence interval. A correlation gives you one parameter. A confidence interval gives you two. Of course the latter will give you more information.
The confidence interval also works well because Z is normally distributed around X. That's very fortunate. Correlations work even when you don't know the distribution. Rework this example, but assume that Z is characterized using mean and stdev but secretly has a heavily-skewed distribution around X, and see how they compare then.
I don't disagree with anything in the above comment. The opening is odd, as others have noted, and I agree. I'm not sure why you're introducing confidence intervals though.
"a variable Z which is experimentally found to have a good chance of lying close to X. Let us suppose that the standard deviation of Z-X is 10% that of X".
So, you have a confidence interval for X around Z.