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, statistical difference or variational distance.

Definition[edit]

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.

Properties[edit]

Relation to other distances[edit]

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

One also has the following inequality, due to Betagnolle and Huber[2] (see, also, Tsybakov[3]), which has the advantage of providing a non-vacuous bound even when :

When the set is countable, the total variation distance is related to the L1 norm by the identity:[4]

The total variation distance is related to the Hellinger distance as follows:[5]

These inequalities follow immediately from the inequalities between the 1-norm and the 2-norm.

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.[6]

See also[edit]

References[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. ^ Bretagnolle, J.; Huber, C, Estimation des densités: risque minimax, Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977), pp. 342–363, Lecture Notes in Math., 649, Springer, Berlin, 1978, Lemma 2.1 (French).
  3. ^ Tsybakov, Alexandre B., Introduction to nonparametric estimation, Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. Springer Series in Statistics. Springer, New York, 2009. xii+214 pp. ISBN 978-0-387-79051-0, Equation 2.25.
  4. ^ David A. Levin, Yuval Peres, Elizabeth L. Wilmer, Markov Chains and Mixing Times, 2nd. rev. ed. (AMS, 2017), Proposition 4.2, p. 48.
  5. ^ Harsha, Prahladh (September 23, 2011). "Lecture notes on communication complexity" (PDF).
  6. ^ 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.