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, statistical difference or variational distance.

Definition

Consider a measurable space $(\Omega, \mathcal)$ and probability measures $P$ and $Q$ defined on $(\Omega, \mathcal)$.
The total variation distance between $P$ and $Q$ is defined as:

:$\delta(P,Q)=\sup_\left, P(A)-Q(A)\.$
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:

:$\delta(P,Q) \le \sqrt.$

One also has the following inequality, due to Bretagnolle and Huber (see, also, Tsybakov), which has the advantage of providing a non-vacuous bound even when $D_(P\parallel Q)>2$:

:$\delta(P,Q) \le \sqrt.$

When $\Omega$ is countable, the total variation distance is related to the L¹ norm by the identity:

:$\delta(P, Q)=\frac12\, P-Q\, _1=\frac12\sum_, P(\)-Q(\),$

The total variation distance is related to the Hellinger distance $H(P,Q)$ as follows:

: $H^2(P,Q) \leq \delta(P,Q) \leq \sqrt 2 H(P,Q).$

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

Connection to transportation theory

The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is $c(x,y) = _$, that is,

:\frac \, P - Q \, _1 = \delta(P,Q) = \inf\ = \inf_\pi \operatorname_

where the expectation is taken with respect to the probability measure $\pi$ on the space where $(x,y)$ lives, and the infimum is taken over all such $\pi$ with marginals $P$ and $Q$, respectively.

See also

*Total variation
*Kolmogorov–Smirnov test
*Wasserstein metric

References

Probability theory
F-divergences

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