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 distances
When the set is countable, the total variation distance is related to the L1 norm by the identity:
Connection to transportation theory
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is , that is,
where the infimum is taken over all probability distributions 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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