Statistical distance

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.

Statistical distance measures are mostly not metrics and they need not be symmetric. Some types of distance measures 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

Metrics

A metric on a set X is a function (called the distance function or simply distance)

d : X × X → R⁺ (where R⁺ is the set of non-negative real numbers). For all x, y, z in X, this function is required to satisfy the following conditions:

d(x, y) ≥ 0 (non-negativity)
d(x, y) = 0 if and only if x = y (identity of indiscernibles. Note that condition 1 and 2 together produce positive definiteness)
d(x, y) = d(y, x) (symmetry)
d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality).

Generalized metrics

Many statistical distances are not metrics, because they lack one or more properties of proper metrics. For example, pseudometrics violate the "positive definiteness" (alternatively, "identity of indescernibles") property (1 & 2 above); quasimetrics violate the symmetry property (3); and semimetrics violate the triangle inequality (4). Statistical distances that satisfy (1) and (2) are referred to as divergences.

Examples

Some important statistical distances include the following:

f-divergence: includes
- Kullback–Leibler divergence
- Hellinger distance
- Total variation distance (sometimes just called "the" statistical distance)
Rényi's divergence
Jensen–Shannon divergence
Lévy–Prokhorov metric
Bhattacharyya distance
Wasserstein metric: also known as the Kantorovich metric, or earth mover's distance
The Kolmogorov–Smirnov statistic represents a distance between two probability distributions defined on a single real variable
The maximum mean discrepancy which is defined in terms of the kernel embedding of distributions
Signal-to-noise ratio distance
Mahalanobis distance
Discriminability index, specifically the Bayes discriminability index is a positive-definite symmetric measure of the overlap of two distributions.
Energy distance
- Distance correlation is a measure of dependence between two random variables, it is zero if and only if the random variables are independent.
The continuous ranked probability score measures how well forecasts that are expressed as probability distributions match observed outcomes. Both the location and spread of the forecast distribution are taken into account in judging how close the distribution is the observed value: see probabilistic forecasting.
Łukaszyk–Karmowski metric is a function defining a distance between two random variables or two random vectors. It does not satisfy the identity of indiscernibles condition of the metric and is zero if and only if both its arguments are certain events described by Dirac delta density probability distribution functions.