Perhaps "Coal Miner's Distance" is an alternative name for "Earth Mover's Distance"? EMD is a metric on probability distributions, also known as the Wasserstein metric. See
https://en.wikipedia.org/wiki/Earth_mover%27s_distance for a description and references.
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Michael W. Trosset
Professor & Chair of Statistics
Executive Director of the
Indiana Statistical Consulting Center
Indiana University, Bloomington, IN
"Luck is probability taken personally.
It is the excitement of bad math."
Penn & Teller
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Is "coal miner's distance" the length of a chord (an imaginary tunnel through the earth) between two points on the surface?