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This question is inspired by the suprisingly complicated Wikipedia page on correlation and dependence.Can you explain distance correlation and brownias covariance as well as the 'Randomized Dependence Coefficient' in lay man's terms and their application, particularly for rationalists? How about the 'correlation ratio', 'polychoric correlation' and 'coefficient of determination'?
A first broad attempt.
The stage is set-up in this way: you observe two set of data, that your model indicates as coming from two distinct sources. The question is: are the two sets related in any way? If so, how much? The 'measure' of such is usually called correlation.
From an objective Bayesian point of view, it doesn't make much sense to talk about correlation between two random variables (it makes no sense to talk about random variable either, but that's another story), because correlation is always model dependent, and probabilities are epistemic. Two ... (read more)