# 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 ${\mathcal {F}}$ of subsets of the sample space $\Omega$ is defined via

$\delta (P,Q)=\sup _{A\in {\mathcal {F}}}\left|P(A)-Q(A)\right|.$ 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:

$\delta (P,Q)\leq {\sqrt {{\frac {1}{2}}D_{\mathrm {KL} }(P\parallel Q)}}.$ When the set is countable, the total variation distance is related to the L1 norm by the identity:

$\delta (P,Q)={\frac {1}{2}}\|P-Q\|_{1}={\frac {1}{2}}\sum _{\omega \in \Omega }|P(\omega )-Q(\omega )|.$ ### Connection to transportation theory

The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is $c(x,y)={\mathbf {1} }_{x\neq y}$ , that is,

${\frac {1}{2}}\|P-Q\|_{1}=\delta (P,Q)=\inf _{\pi }\operatorname {E} _{\pi }[{\mathbf {1} }_{x\neq y}],$ 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.