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In [[probability theory]], an '''''ƒ''-divergence''' is a function ''D''<sub>''f''</sub>&thinsp;(''P''&thinsp;&thinsp;||&thinsp;''Q'') that measures the difference between two [[probability distributions]] ''P'' and ''Q''. It helps the intuition to think of the divergence as an average, weighted by the function ''f'', of the [[odds ratio]] given by ''P'' and ''Q''.
 
These divergences were introduced and studied independently by {{harvtxt|Csiszár|1963}}, {{harvtxt|Morimoto|1963}} and {{harvtxt|Ali|Silvey|1966}} and are sometimes known as Csiszár ''ƒ''-divergences, Csiszár-Morimoto divergences or Ali-Silvey distances.
 
== Definition ==
Let ''P'' and ''Q'' be two probability distributions over a space &Omega; such that ''P'' is [[Absolute_continuity#Absolute_continuity_of_measures|absolutely continuous]] with respect to ''Q''. Then, for a [[convex function]] ''f'' such that ''f''(1) = 0, the ''f''-divergence of ''Q'' from ''P'' is defined as
 
:<math> D_f(P\parallel Q) \equiv \int_{\Omega} f\left(\frac{dP}{dQ}\right)\,dQ.</math>
 
If ''P'' and ''Q'' are both absolutely continuous with respect to a reference distribution ''μ'' on &Omega; then their  [[probability density function|probability densities]] ''p'' and ''q'' satisfy ''dP = p dμ'' and ''dQ = q dμ''. In this case the ''f''-divergence can be written as
 
:<math> D_f(P\parallel Q) = \int_{\Omega} f\left(\frac{p(x)}{q(x)}\right)q(x)\,d\mu(x).</math>
 
The f-divergences can be expressed using Taylor series and rewritten using a weighted sum of chi-type distances ({{harvtxt|Nielsen|Nock|2013}}.
 
== Instances of ''f''-divergences ==
Many common divergences, such as [[KL-divergence]], [[Hellinger distance]], and [[total variation distance]], are special cases of ''f''-divergence, coinciding with a particular choice of ''f''. The following table lists many of the common divergences between probability distributions and the ''f'' function to which they correspond (cf. {{harvtxt|Liese|Vajda|2006}}).
 
{| class="wikitable"
|-
! Divergence
! Corresponding ''f(t)''
|-
| [[KL-divergence]]
| <math> t \ln t \, , -\ln t</math>
|-
| [[Hellinger distance]]
| <math>(\sqrt{t} - 1)^2,\,2(1-\sqrt{t})</math>
|-
| [[Total variation distance]]
| <math>|t - 1| \,</math>
|-
| <math> \chi^2</math>-divergence
| <math>(t - 1)^2,\,t^2 -1</math>
|-
| ''α''-divergence
| <math>\begin{cases}
    \frac{4}{1-\alpha^2}\big(1 - t^{(1+\alpha)/2}\big), & \text{if}\ \alpha\neq\pm1, \\
    t \ln t, & \text{if}\ \alpha=1, \\
    - \ln t, & \text{if}\ \alpha=-1
  \end{cases}</math>
|}
 
== Properties ==
{{unordered list
|1= '''Non-negativity''': the ''ƒ''-divergence is always positive; it's zero if and only if the measures ''P'' and ''Q'' coincide. This follows immediately from [[Jensen’s inequality]]:
: <math>
    D_f(P\!\parallel\!Q) = \int \!f\bigg(\frac{dP}{dQ}\bigg)dQ \geq f\bigg( \int\frac{dP}{dQ}dQ\bigg) = f(1) = 0.
  </math>
 
|2= '''Monotonicity''': if ''κ'' is an arbitrary [[transition probability]] that transforms measures ''P'' and ''Q'' into ''P''<sub>''κ''</sub> and ''Q''<sub>''κ''</sub> correspondingly, then
: <math>
    D_f(P\!\parallel\!Q) \geq D_f(P_\kappa\!\parallel\!Q_\kappa).
  </math>
The equality here holds if and only if the transition is induced from a [[sufficient statistic]] with respect to {''P'', ''Q''}.
 
|3=  '''Joint Convexity''': for any {{nowrap|0 ≤ ''λ'' ≤ 1}}
: <math>
    D_f\Big(\lambda P_1 + (1-\lambda)P_2 \parallel \lambda Q_1 + (1-\lambda)Q_2\Big) \leq \lambda D_f(P_1\!\parallel\!Q_1) + (1-\lambda)D_f(P_2\!\parallel\!Q_2).
  </math>
This follows from the convexity of the mapping <math>(p,q) \mapsto q f(p/q)</math> on <math>\mathbb{R}_+^2</math>.
}}
 
== References ==
{{refbegin}}
* {{cite journal
  | first = I.
  | last = Csiszár
  | year = 1963
  | title = Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten
  | journal = Magyar. Tud. Akad. Mat. Kutato Int. Kozl
  | volume = 8
  | pages = 85–108
  | ref = CITEREFCsisz.C3.A1r1963
  }}
* {{cite journal
  | doi = 10.1143/JPSJ.18.328
  | first = T.
  | last = Morimoto
  | year = 1963
  | title = Markov processes and the H-theorem
  | journal = J. Phys. Soc. Jap.
  | volume = 18
  | issue = 3
  | pages = 328–331
  | ref = CITEREFMorimoto1963
  }}
* {{cite journal
  | first1 = S. M. | last1 = Ali
  | first2 = S. D. | last2 = Silvey
  | year = 1966
  | title = A general class of coefficients of divergence of one distribution from another
  | journal = [[Journal of the Royal Statistical Society]], Series B
  | volume = 28
  | issue = 1
  | pages = 131–142
  | jstor = 2984279 | mr = 0196777
  | ref = CITEREFAliSilvey1966
  }}
* {{cite journal
  | first = I.
  | last = Csiszár
  | year = 1967
  | title = Information-type measures of difference of probability distributions and indirect observation
  | journal = Studia Scientiarum Mathematicarum Hungarica
  | volume = 2
  | pages = 229–318
  | ref = CITEREFCsisz.C3.A1r1967
  }}
* {{cite journal
  | first1 = I. | last1 = Csiszár | authorlink1 = Imre Csiszár
  | first2 = P. | last2 = Shields
  | year = 2004
  | title = Information Theory and Statistics: A Tutorial
  | journal = Foundations and Trends in Communications and Information Theory
  | volume = 1
  | issue = 4
  | pages = 417–528
  | doi = 10.1561/0100000004
  | url = http://www.renyi.hu/~csiszar/Publications/Information_Theory_and_Statistics:_A_Tutorial.pdf   
  | accessdate = 2009-04-08
  }}
* {{cite journal
  | first1 = F. | last1 = Liese
  | first2 = I. | last2 = Vajda
  | year = 2006
  | title = On divergences and informations in statistics and information theory
  | journal = [[IEEE Transactions on Information Theory]]
  | volume = 52
  | issue = 10
  | pages = 4394–4412
  | doi = 10.1109/TIT.2006.881731
  | ref = CITEREFLieseVajda2006
  }}
* {{cite journal
  | first1 = F. | last1 = Nielsen
  | first2 = R. | last2 = Nock
  | year = 2013
  | title = On the Chi square and higher-order Chi distances for approximating f-divergences
  | journal = [[arXiv]]
  | volume = 1309.3029
| url = http://arxiv.org/abs/1309.3029
  | ref = CITEREFNielsenNock2013
  }}
* {{cite arxiv
  | first1 = J-F. | last1 = Coeurjolly
  | first2 = R. | last2 = Drouihet
  | year = 2006
  | title = Normalized information-based divergences
  | journal = | eprint = math/0604246
  | ref = arXiv:math/0604246
  }}
{{refend}}
 
[[Category:F-divergences| ]]

Latest revision as of 04:14, 18 April 2014

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