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RichardKennaway comments on Causality does not imply correlation - Less Wrong
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Comments (54)
gjm has read the note I linked; I suggest you do the same. That is what a link is for.
Not particularly. The speed of a car, the temperature of a room, the height of an aircraft: such things are all around you. Stating the property of the whole real line is an idealisation, but Theorem 1 of the note treats of finite intervals also, and there is a version of the theorems for time series.
In keeping with the terminology established at the note I linked, I take this to mean x=t. Yes, it is not true of x=t. This does not have an average over the whole real line.
Full disclosure: actually I didn't, I just inferred what the notation had to mean :-).
I wish I hadn't made my comment about precision, which was too nitpicking and unhelpful. But as long as we're being snippy with each other:
To be excruciatingly precise: You just said you were being precise, then said "Let x be a differentiable real function." That isn't precise; you need to specify right there that it's a function of t. If you'd said the link stated it precisely, that would be different.
I admit that I would have interpreted it correctly by making the most-favorable, most-reasonable interpretation and assuming x was a function of t. But, because of the sorts of things I usually see done with x and t, I assumed that x was a function of time, and the function of interest was some function of x(t), and I jumped to the conclusion that you meant to say "Let f(x) be a differentiable real function." Which I would not have done had you in fact been precise, and said "Let x(t) be a differentiable real function."