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.

Definition

The total variation distance between two probability measures P and Q on a sigma-algebra of subsets of the sample space is defined via
Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.

Properties

Relation to other distances

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:

Connection to transportation theory">Transportation theory (mathematics)">transportation theory

The total variation distance 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.