F-divergence

In probability theory, an ƒ-divergence is a function D_f (P  || 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 by Alfréd Rényi

in the same paper where he introduced the well-known Rényi entropy. He proved that these divergences decrease  in Markov Processes. f-divergences were studied further   independently by Csiszár (1963), Morimoto (1963) and 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 Ω such that P is absolutely continuous with respect to Q. Then, for a convex function f such that f(1) = 0, the f-divergence of P from Q is defined as

$D_{f}(P\parallel Q)\equiv \int _{\Omega }f\left({\frac {dP}{dQ}}\right)\,dQ.$

If P and Q are both absolutely continuous with respect to a reference distribution μ on Ω then their probability densities p and q satisfy dP = p dμ and dQ = q dμ. In this case the f-divergence can be written as

$D_{f}(P\parallel Q)=\int _{\Omega }f\left({\frac {p(x)}{q(x)}}\right)q(x)\,d\mu (x).$

The f-divergences can be expressed using Taylor series and rewritten using a weighted sum of chi-type distances (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. Liese & Vajda (2006)).

Divergence	Corresponding f(t)
KL-divergence	$t\log t$
reverse KL-divergence	$-\log t$
squared Hellinger distance	$({\sqrt {t}}-1)^{2},\,2(1-{\sqrt {t}})$
Total variation distance	${\frac {1}{2}}|t-1|\,$
Pearson $\chi ^{2}$-divergence	$(t-1)^{2},\,t^{2}-1,\,t^{2}-t$
Neyman $\chi ^{2}$-divergence (reverse Pearson)	${\frac {1}{t}}-1,\,{\frac {1}{t}}-t$
α-divergence	${\begin{cases}{\frac {4}{1-\alpha ^{2}}}{\big (}1-t^{(1+\alpha )/2}{\big )},&{\text{if}}\ \alpha \neq \pm 1,\\t\ln t,&{\text{if}}\ \alpha =1,\\-\ln t,&{\text{if}}\ \alpha =-1\end{cases}}$
α-divergence (other designation)	${\begin{cases}{\frac {t^{\alpha }-t}{\alpha (\alpha -1)}},&{\text{if}}\ \alpha \neq 0,\,\alpha \neq 1,\\t\ln t,&{\text{if}}\ \alpha =1,\\-\ln t,&{\text{if}}\ \alpha =0\end{cases}}$

The function $f(t)$ is defined up to the summand $c(t-1)$, where $c$ is any constant.

Properties

• 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:

  $D_{f}(P\!\parallel \!Q)=\int \!f{\bigg (}{\frac {dP}{dQ}}{\bigg )}dQ\geq f{\bigg (}\int {\frac {dP}{dQ}}dQ{\bigg )}=f(1)=0.$

• Monotonicity: if κ is an arbitrary transition probability that transforms measures P and Q into P_κ and Q_κ correspondingly, then
  $D_{f}(P\!\parallel \!Q)\geq D_{f}(P_{\kappa }\!\parallel \!Q_{\kappa }).$
  The equality here holds if and only if the transition is induced from a sufficient statistic with respect to {P, Q}.
• Joint Convexity: for any 0 ≤ λ ≤ 1

  $D_{f}{\Big (}\lambda P_{1}+(1-\lambda )P_{2}\parallel \lambda Q_{1}+(1-\lambda )Q_{2}{\Big )}\leq \lambda D_{f}(P_{1}\!\parallel \!Q_{1})+(1-\lambda )D_{f}(P_{2}\!\parallel \!Q_{2}).$

  This follows from the convexity of the mapping $(p,q)\mapsto qf(p/q)$ on $\mathbb {R} _{+}^{2}$.

In particular, the monotonicity implies that if a Markov process has a positive equilibrium probability distribution $P^{*}$ then $D_{f}(P(t)\parallel P^{*})$ is a monotonic (non-increasing) function of time, where the probability distribution $P(t)$ is a solution of the Kolmogorov forward equations (or Master equation), used to describe the time evolution of the probability distribution in the Markov process. This means that all f-divergences $D_{f}(P(t)\parallel P^{*})$ are the Lyapunov functions of the Kolmogorov forward equations. Reverse statement is also true: If $H(P)$ is a Lyapunov function for all Markov chains with positive equilibrium $P^{*}$ and is of the trace-form ($H(P)=\sum _{i}f(P_{i},P_{i}^{*})$) then $H(P)=D_{f}(P(t)\parallel P^{*})$, for some convex function f. For example, Bregman divergences in general do not have such property and can increase in Markov processes.

See also

• Kullback–Leibler divergence
• Bregman divergence

References

• Csiszár, I. (1963). "Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten". Magyar. Tud. Akad. Mat. Kutato Int. Kozl. 8: 85–108.{{cite journal}}: CS1 maint: ref duplicates default (link)
• Morimoto, T. (1963). "Markov processes and the H-theorem". J. Phys. Soc. Jpn. 18 (3): 328–331. Bibcode:1963JPSJ...18..328M. doi:10.1143/JPSJ.18.328.{{cite journal}}: CS1 maint: ref duplicates default (link)
• Ali, S. M.; Silvey, S. D. (1966). "A general class of coefficients of divergence of one distribution from another". Journal of the Royal Statistical Society, Series B. 28 (1): 131–142. JSTOR 2984279. MR 0196777.{{cite journal}}: CS1 maint: ref duplicates default (link)
• Csiszár, I. (1967). "Information-type measures of difference of probability distributions and indirect observation". Studia Scientiarum Mathematicarum Hungarica. 2: 229–318.
• Csiszár, I.; Shields, P. (2004). "Information Theory and Statistics: A Tutorial" (PDF). Foundations and Trends in Communications and Information Theory. 1 (4): 417–528. doi:10.1561/0100000004. Retrieved 2009-04-08.
• Liese, F.; Vajda, I. (2006). "On divergences and informations in statistics and information theory". IEEE Transactions on Information Theory. 52 (10): 4394–4412. doi:10.1109/TIT.2006.881731.{{cite journal}}: CS1 maint: ref duplicates default (link)
• Nielsen, F.; Nock, R. (2013). "On the Chi square and higher-order Chi distances for approximating f-divergences". IEEE Signal Processing Letters. 21: 10–13. arXiv:1309.3029. Bibcode:2014ISPL...21...10N. doi:10.1109/LSP.2013.2288355.{{cite journal}}: CS1 maint: ref duplicates default (link)
• Coeurjolly, J-F.; Drouilhet, R. (2006). "Normalized information-based divergences". arXiv:math/0604246.