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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'''.
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| ==Definition==
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| The total variation distance between two [[probability measure]]s ''P'' and ''Q'' on a [[sigma-algebra]] ''<math>\mathcal{F}</math>'' of [[subset]]s of the sample space <math>\Omega</math> is defined via<ref name=Chatterjee2007>{{cite web|last=Chatterjee|first=Sourav|title=Distances between probability measures|url=http://www.stat.berkeley.edu/~sourav/Lecture2.pdf|publisher=UC Berkeley|accessdate=21 June 2013}}</ref>
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| :<math>\delta(P,Q)=\sup_{ A\in \mathcal{F}}\left|P(A)-Q(A)\right|. </math>
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| Informally, this is the largest possible difference between the probabilities that the two [[probability distribution]]s can assign to the same event.
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| For a [[Categorical distribution|finite alphabet]] we can relate the total variation distance to the [[Lp-norm|1-norm]] of the difference of the two probability distributions as follows:<ref>http://books.google.com/books?id=6Cg5Nq5sSv4C&lpg=PP1&pg=PA48#v=onepage&q&f=false</ref>
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| :<math>\delta(P,Q) = \frac 1 2 \|P-Q\|_1 = \frac 1 2 \sum_x \left| P(x) - Q(x) \right|\;.</math>
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| For arbitrary sample spaces, an equivalent definition of the total variation distance is
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| :<math>\delta(P,Q) = \frac 1 2 \int_\Omega \left| f_P - f_Q \right|d\mu\;.</math>
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| where <math>\mu</math> is an arbitrary positive measure such that both <math>P</math> and <math>Q</math> are [[absolutely continuous]] with respect to it and where <math>f_P</math> and <math>f_Q</math> are the [[Radon-Nikodym]] derivatives of <math>P</math> and <math>Q</math> with respect to <math>\mu</math>.
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| The total variation distance is related to the [[Kullback–Leibler divergence]] by [[Pinsker's inequality]].
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| ==See also==
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| *[[Total variation]]
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| *[[Kolmogorov–Smirnov test]]
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| *[[Wasserstein metric]]
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| ==References==
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| {{reflist}}
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| [[Category:Probability theory]]
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| [[Category:F-divergences]]
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| {{probability-stub}}
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Latest revision as of 02:17, 5 May 2014
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