Talk:Probability metric

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To the best of my knowledge (and I'm not a probabilist, so my understanding of the nuances of this terminology isn't rock solid) a 'probability metric' is a metric on a space of probability functions. Not on a space of random variables. But regardless of the nomenclature, this page seems to define "probability metric" as the L_1 distance between two random variables under sort of vague and varying unspecified distribution functions. Which is weird. When I have time in the future, I'll try to edit it, but maybe a probability grad student could/should clean it up? Gray 23:59, 2 September 2007 (UTC)

Dear Gray, The 'probability metric' is not an L1 distance of two random variables, as it covers also random vectors. The distribution functions are arbitrary but it is not vague as it is a consequence of the fact that a given distribution is the property of the random variable or vector which distance from another variable or vector is measured. After specifying distributions and integrating the PM (which may be complicated e.g. for random vectors of different distributions of coefficients) you obtain a given, particular form of the probability metric. Examples of particular PM for two continuous random variables with normal (NN) and rectangular distributions (RR) are shown.--Guswen 17:31, 3 September 2007 (UTC)

As written in the article, it is vague. The notation D_z,z is used without being introduced, and is apparently redundant. I also don't see how to reconcile the total variation (for example) with this definition since the article seems to treat the joint distribution of X and Y fixed as well as their marginals. Isn't the total variation a semi-metric on the space of probability measures? Gray 04:30, 4 September 2007 (UTC)

I introduced a brief explanation of the subscript notation. I agree that the proposed notation may need some improvements in case of joint probability density function (F(x, y)) of two dependent random variables or vectors. Yet it explains which distributions were taken as PM arguments. PM is not a semi-metric as it satisfies the triangle inequality condition. --Guswen 18:14, 4 September 2007 (UTC)