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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 just called "the" statistical 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.

For a finite alphabet we can relate the total variation distance to the 1-norm of the difference of the two probability distributions as follows:

$\delta (P,Q)={\frac {1}{2}}\|P-Q\|_{1}={\frac {1}{2}}\sum _{x}\left|P(x)-Q(x)\right|\;.$

For arbitrary sample spaces, an equivalent definition of the total variation distance is

$\delta (P,Q)={\frac {1}{2}}\int _{\Omega }\left|f_{P}-f_{Q}\right|d\mu \;.$

where $\mu$ is an arbitrary positive measure such that both $P$ and $Q$ are absolutely continuous with respect to it and where $f_{P}$ and $f_{Q}$ are the Radon-Nikodym derivatives of $P$ and $Q$ with respect to $\mu$.

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

See also

• Total variation
• Kolmogorov–Smirnov test
• Wasserstein metric

References

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