Total variation distance of probability measures

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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.


The total variation distance between two probability measures P and Q on a sigma-algebra of subsets of the sample space is defined via[1]

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[edit]

The total variation distance is related to the Kullback–Leibler divergence by Pinsker's inequality:

When the set is countable, the total variation distance is related to the L1 norm by the identity:[2]

Connection to transportation theory[edit]

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[3].

See also[edit]


  1. ^ Chatterjee, Sourav. "Distances between probability measures" (PDF). UC Berkeley. Archived from the original (PDF) on July 8, 2008. Retrieved 21 June 2013.
  2. ^ David A. Levin, Yuval Peres, Elizabeth L. Wilmer, 'Markov Chains and Mixing Times', 2nd. rev. ed. (AMS, 2017), Proposition 4.2, p. 48.
  3. ^ 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.