Total variation distance

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 just called "the" statistical distance.

The total variation distance between two probability measures P and Q on a sigma-algebra ${\displaystyle {\mathcal {F}}}$ of subsets of the sample space ${\displaystyle \Omega }$ is defined via

Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.

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

On a finite probability space, the total variation distance is related to the L¹ norm by the identity:

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

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