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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 $\mathrm{\Omega }$ is defined via

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

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

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:

$\delta (P,Q)=\frac{1}{2}\Vert P-Q{\Vert }_1=\frac{1}{2}\sum _x|P(x)-Q(x)|.$

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

$\delta (P,Q)=\frac{1}{2}\underset{\mathrm{\Omega }}{\int }|f_P-f_Q|d\mu .$

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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