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 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^[1]

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

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

Special cases

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:^[2]

$\delta(P,Q) = \frac 1 2 \|P-Q\|_1 = \frac 1 2 \sum_x \left| P(x) - Q(x) \right|\;.$

Similarly, for arbitrary sample space $\Omega$, measure $\mu$, and probability measures $P$ and $Q$ with Radon-Nikodym derivatives $f_P$ and $f_Q$ with respect to $\mu$, an equivalent definition of the total variation distance is

$\delta(P,Q) = \frac{1}{2} \|f_P-f_Q\|_{L_1(\mu)} = \frac 1 2 \int_\Omega \left| f_P - f_Q \right|d\mu\;.$

Relationship with other concepts

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

1. Chatterjee, Sourav. "Distances between probability measures" (PDF). UC Berkeley. Archived from the original (PDF) on July 8, 2008. Retrieved 21 June 2013.
2. Levin, David Asher; Peres, Yuval; Wilmer, Elizabeth Lee. Markov Chains and Mixing Times. American Mathematical Soc. ISBN 9780821886274.