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 just called "the" statistical 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
On a finite probability space, the total variation distance is related to the L1 norm by the identity:
Connection to Transportation theory
The total variation distance arises as twice 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', Proposition 5.2, p.50
- Villani, Cédric (2009). Optimal Transport, Old and New. Springer-Verlag Berlin Heidelberg. p. 22. ISBN 978-3-540-71049-3. doi:10.1007/978-3-540-71050-9.
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