Total variation distance of probability measures
In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance or variational distance.
Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.
Relation to other distancesEdit
Connection to transportation theoryEdit
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is , that is,
where the expectation is taken with respect to the probability measure on the space where lives, and the infimum is taken over all such with marginals and , respectively.
- Chatterjee, Sourav. "Distances between probability measures" (PDF). UC Berkeley. Archived from the original (PDF) on July 8, 2008. Retrieved 21 June 2013.
- David A. Levin, Yuval Peres, Elizabeth L. Wilmer, 'Markov Chains and Mixing Times', 2nd. rev. ed. (AMS, 2017), Proposition 4.2, p. 48.
- Villani, Cédric (2009). Optimal Transport, Old and New. Grundlehren der mathematischen Wissenschaften. 338. Springer-Verlag Berlin Heidelberg. p. 10. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3.
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