In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be between an individual sample point and a population or a wider sample of points.

A distance between populations can be interpreted as measuring the distance between two probability distributions and hence they are essentially measures of distances between probability measures. Where statistical distance measures relate to the differences between random variables, these may have statistical dependence,^[1] and hence these distances are not directly related to measures of distances between probability measures. Again, a measure of distance between random variables may relate to the extent of dependence between them, rather than to their individual values.

Many statistical distance measures are not metrics, and some are not symmetric. Some types of distance measures, which generalize squared distance, are referred to as (statistical) divergences.

Terminology

Many terms are used to refer to various notions of distance; these are often confusingly similar, and may be used inconsistently between authors and over time, either loosely or with precise technical meaning. In addition to "distance", similar terms include deviance, deviation, discrepancy, discrimination, and divergence, as well as others such as contrast function and metric. Terms from information theory include cross entropy, relative entropy, discrimination information, and information gain.

Distances as metrics

Statistically close

The total variation distance of two distributions ${\displaystyle X}$ and ${\displaystyle Y}$ over a finite domain ${\displaystyle D}$, (often referred to as statistical difference^[2] or statistical distance^[3] in cryptography) is defined as

${\displaystyle \Delta (X,Y)={\frac {1}{2}}\sum _{\alpha \in D}|\Pr[X=\alpha ]-\Pr[Y=\alpha ]|}$.

We say that two probability ensembles ${\displaystyle \{X_{k}\}_{k\in \mathbb {N} }}$ and ${\displaystyle \{Y_{k}\}_{k\in \mathbb {N} }}$ are statistically close if ${\displaystyle \Delta (X_{k},Y_{k})}$ is a negligible function in ${\displaystyle k}$.

Examples

See also

• Probabilistic metric space
• Randomness extractor
• Similarity measure
• Zero-knowledge proof

Notes

External links

• Distance and Similarity Measures (Wolfram Alpha)

References

• Dodge, Y. (2003) Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9