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 and studied 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 Q from P 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	$t \ln t \, , -\ln t$
Hellinger distance	$(\sqrt{t} - 1)^2,\,2(1-\sqrt{t})$
Total variation distance	$\frac12|t - 1| \,$
$\chi^2$-divergence	$(t - 1)^2,\,t^2 -1$
α-divergence	$\begin{cases} \frac{4}{1-\alpha^2}\big(1 - t^{(1+\alpha)/2}\big), & \text{if}\ \alpha\neq\pm1, \\ t \ln t, & \text{if}\ \alpha=1, \\ - \ln t, & \text{if}\ \alpha=-1 \end{cases}$

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 $(p,q) \mapsto q f(p/q)$ on $\mathbb{R}_+^2$.

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.
• Morimoto, T. (1963). "Markov processes and the H-theorem". J. Phys. Soc. Jap. 18 (3): 328–331. doi:10.1143/JPSJ.18.328.
• 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.
• 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.
• Nielsen, F.; Nock, R. (2013). "On the Chi square and higher-order Chi distances for approximating f-divergences". arXiv. 1309.3029.
• Coeurjolly, J-F.; Drouilhet, R. (2006). "Normalized information-based divergences". arXiv:math/0604246.