F-divergence

Definition

Non-singular case

Let ${\displaystyle P}$ and ${\displaystyle Q}$ be two probability distributions over a space ${\displaystyle \mathrm{\Omega }}$, such that ${\displaystyle P\ll Q}$, that is, ${\displaystyle P}$ is absolutely continuous with respect to ${\displaystyle Q}$ (meaning ${\displaystyle Q>0}$ wherever ${\displaystyle P>0}$). Then, for a convex function ${\displaystyle f:[0,+\mathrm{\infty })\to (-\mathrm{\infty },+\mathrm{\infty }]}$ such that ${\displaystyle f(x)}$ is finite for all ${\displaystyle x>0}$, ${\displaystyle f(1)=0}$, and ${\displaystyle f(0)=\underset{t\to 0^{+}}{lim}f(t)}$ (which could be infinite), the ${\displaystyle f}$-divergence of ${\displaystyle P}$ from ${\displaystyle Q}$ is defined as

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

We call ${\displaystyle f}$ the generator of ${\displaystyle D_f}$.

In concrete applications, there is usually a reference distribution ${\displaystyle \mu }$ on ${\displaystyle \mathrm{\Omega }}$ (for example, when ${\displaystyle \mathrm{\Omega }={\mathbb{R}}^n}$, the reference distribution is the Lebesgue measure), such that ${\displaystyle P,Q\ll \mu }$, then we can use Radon–Nikodym theorem to take their probability densities ${\displaystyle p}$ and ${\displaystyle q}$, giving

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

When there is no such reference distribution ready at hand, we can simply define ${\displaystyle \mu =P+Q}$, and proceed as above. This is a useful technique in more abstract proofs.

Extension to singular measures

The above definition can be extended to cases where ${\displaystyle P\ll Q}$ is no longer satisfied (Definition 7.1 of ).

Since ${\displaystyle f}$ is convex, and ${\displaystyle f(1)=0}$ , the function ${\displaystyle \frac{f(x)}{x-1}}$ must be nondecreasing, so there exists ${\displaystyle f^{\prime }(\mathrm{\infty }):=\underset{x\to \mathrm{\infty }}{lim}f(x)/x}$, taking value in ${\displaystyle (-\mathrm{\infty },+\mathrm{\infty }]}$.

Since for any ${\displaystyle p(x)>0}$, we have ${\displaystyle \underset{q(x)\to 0}{lim}q(x)f\left(\frac{p(x)}{q(x)}\right)=p(x)f^{\prime }(\mathrm{\infty })}$ , we can extend f-divergence to the ${\displaystyle P\ll <!--\; --≯Q}$.

Properties

Basic relations between f-divergences

• Linearity: ${\displaystyle D_{\sum _i{a}_i{f}_i}=\sum _i{a}_i{D}_{f_i}}$ given a finite sequence of nonnegative real numbers ${\displaystyle a_i}$ and generators ${\displaystyle f_i}$.

• ${\displaystyle D_f=D_g}$ iff ${\displaystyle f(x)=g(x)+c(x-1)}$ for some ${\displaystyle c\in \mathbb{R}}$.

Proof

If ${\displaystyle f(x)=g(x)+c(x-1)}$, then ${\displaystyle D_f=D_g}$ by definition.

Conversely, if ${\displaystyle D_f-D_g=0}$, then let ${\displaystyle h=f-g}$. For any two probability measures ${\displaystyle P,Q}$ on the set ${\displaystyle \{0,1\}}$, since ${\displaystyle D_f(P\Vert Q)-D_g(P\Vert Q)=0}$, we get ${\displaystyle h(P_1/Q_1)=-\frac{Q_0}{Q_1}h(P_0/Q_0)}$

Since each probability measure ${\displaystyle P,Q}$ has one degree of freedom, we can solve ${\displaystyle \frac{P_0}{Q_0}=a,\frac{P_1}{Q_1}=x}$ for every choice of ${\displaystyle 0<a<1<x}$.

Linear algebra yields ${\displaystyle Q_0=\frac{x-1}{x-a},Q_1=\frac{1-a}{x-a}}$, which is a valid probability measure. Then we obtain ${\displaystyle h(x)=\frac{h(a)}{a-1}(x-1),h(a)=\frac{h(x)}{x-1}(a-1)}$.

Thus ${\displaystyle h(x)=\{\begin{array}{l}{c}_1(x-1)\text{if }x>1,\\ c_0(x-1)\text{if }0<x<1,\end{array}}$ for some constants ${\displaystyle c_0,c_1}$. Plugging the formula into ${\displaystyle h(x)=\frac{h(a)}{a-1}(x-1)}$ yields ${\displaystyle c_0=c_1}$.

Basic properties of f-divergences

• Non-negativity: the ƒ-divergence is always positive; it is zero if the measures P and Q coincide. This follows immediately from Jensen’s inequality:

  ${\displaystyle D_f(P\parallel Q)=\int f(\frac{dP}{dQ})dQ\ge f(\int \frac{dP}{dQ}dQ)=f(1)=0.}$

• Data-processing inequality: if κ is an arbitrary transition probability that transforms measures P and Q into P_κ and Q_κ correspondingly, then

  ${\displaystyle D_f(P\parallel Q)\ge D_f(P_{\kappa }\parallel Q_{\kappa }).}$

  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,

  ${\displaystyle D_f(\lambda P_1+(1-\lambda )P_2\parallel \lambda Q_1+(1-\lambda )Q_2)\le \lambda D_f(P_1\parallel Q_1)+(1-\lambda )D_f(P_2\parallel Q_2).}$

  This follows from the convexity of the mapping ${\displaystyle (p,q)\mapsto qf(p/q)}$ on ${\displaystyle {\mathbb{R}}_{+}^2}$.
• Reversal by convex inversion: for any function ${\displaystyle f}$, its convex inversion is defined as ${\displaystyle g(t):=tf(1/t)}$. When ${\displaystyle f}$ satisfies the defining features of a f-divergence generator (${\displaystyle f(x)}$ is finite for all ${\displaystyle x>0}$, ${\displaystyle f(1)=0}$, and ${\displaystyle f(0)=\underset{t\to 0^{+}}{lim}f(t)}$), then ${\displaystyle g}$ satisfies the same features, and thus defines a f-divergence ${\displaystyle D_g}$. This is the "reverse" of ${\displaystyle D_f}$, in the sense that ${\displaystyle D_g(P\Vert Q)=D_f(Q\Vert P)}$ for all ${\displaystyle P,Q}$ that are absolutely continuous with respect to each other. In this way, every f-divergence ${\displaystyle D_f}$ can be turned symmetric by ${\displaystyle D_{\frac{1}{2}(f+g)}}$. For example, performing this symmetrization turns KL-divergence into Jeffreys divergence.

In particular, the monotonicity implies that if a Markov process has a positive equilibrium probability distribution ${\displaystyle P^{\ast }}$ then ${\displaystyle D_f(P(t)\parallel P^{\ast })}$ is a monotonic (non-increasing) function of time, where the probability distribution ${\displaystyle P(t)}$ 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 ${\displaystyle D_f(P(t)\parallel P^{\ast })}$ are the Lyapunov functions of the Kolmogorov forward equations. The converse statement is also true: If ${\displaystyle H(P)}$ is a Lyapunov function for all Markov chains with positive equilibrium ${\displaystyle P^{\ast }}$ and is of the trace-form (${\displaystyle H(P)=\sum _{i}f(P_i,P_i^{\ast })}$) then ${\displaystyle H(P)=D_f(P(t)\parallel P^{\ast })}$, for some convex function f. For example, Bregman divergences in general do not have such property and can increase in Markov processes.

Analytic properties

The f-divergences can be expressed using Taylor series and rewritten using a weighted sum of chi-type distances (Nielsen & Nock (2013)).

Basic variational representation

Let ${\displaystyle f^{\ast }}$ be the convex conjugate of ${\displaystyle f}$. Let ${\displaystyle \mathrm{e}\mathrm{f}\mathrm{f}\mathrm{d}\mathrm{o}\mathrm{m}(f^{\ast })}$ be the effective domain of ${\displaystyle f^{\ast }}$, that is, ${\displaystyle \mathrm{e}\mathrm{f}\mathrm{f}\mathrm{d}\mathrm{o}\mathrm{m}(f^{\ast })=\{y:f^{\ast }(y)<\mathrm{\infty }\}}$. Then we have two variational representations of ${\displaystyle D_f}$, which we describe below.

Under the above setup,

Theorem—${\displaystyle D_f(P;Q)=\underset{g:\mathrm{\Omega }\to \mathrm{e}\mathrm{f}\mathrm{f}\mathrm{d}\mathrm{o}\mathrm{m}(f^{\ast })}{sup}{E}_P[g]-E_Q[f^{\ast }\circ g]}$.

This is Theorem 7.24 in.

Example applications

Using this theorem on total variation distance, with generator ${\displaystyle f(x)=\frac{1}{2}|x-1|,}$ its convex conjugate is ${\displaystyle f^{\ast }(x^{\ast })=\{\begin{array}{l}{x}^{\ast }\text{ on }[-1/2,1/2],\\ +\mathrm{\infty }\text{ else.}\end{array}}$, and we obtain $${\displaystyle TV(P\Vert Q)=\underset{|g|\le 1/2}{sup}{E}_P[g(X)]-E_Q[g(X)].}$$ For chi-squared divergence, defined by ${\displaystyle f(x)=(x-1{)}^2,f^{\ast }(y)=y^2/4+y}$, we obtain $${\displaystyle {\chi }^2(P;Q)=\underset{g}{sup}{E}_P[g(X)]-E_Q[g(X{)}^2/4+g(X)].}$$ Since the variation term is not affine-invariant in ${\displaystyle g}$, even though the domain over which ${\displaystyle g}$ varies is affine-invariant, we can use up the affine-invariance to obtain a leaner expression.

Replacing ${\displaystyle g}$ by ${\displaystyle ag+b}$ and taking the maximum over ${\displaystyle a,b\in \mathbb{R}}$, we obtain $${\displaystyle {\chi }^2(P;Q)=\underset{g}{sup}\frac{(E_P[g(X)]-E_Q[g(X)]{)}^2}{Va{r}_Q[g(X)]},}$$ which is just a few steps away from the Hammersley–Chapman–Robbins bound and the Cramér–Rao bound (Theorem 29.1 and its corollary in ).

For ${\displaystyle \alpha }$-divergence with ${\displaystyle \alpha \in (-\mathrm{\infty },0)\cup (0,1)}$, we have ${\displaystyle f_{\alpha }(x)=\frac{x^{\alpha }-\alpha x-(1-\alpha )}{\alpha (\alpha -1)}}$, with range ${\displaystyle x\in [0,\mathrm{\infty })}$. Its convex conjugate is ${\displaystyle f_{\alpha }^{\ast }(y)=\frac{1}{\alpha }(x(y{)}^{\alpha }-1)}$ with range ${\displaystyle y\in (-\mathrm{\infty },(1-\alpha {)}^{-1})}$, where ${\displaystyle x(y)=((\alpha -1)y+1{)}^{\frac{1}{\alpha -1}}}$.

Applying this theorem yields, after substitution with ${\displaystyle h=((\alpha -1)g+1{)}^{\frac{1}{\alpha -1}}}$, $${\displaystyle D_{\alpha }(P\Vert Q)=\frac{1}{\alpha (1-\alpha )}-\underset{h:\mathrm{\Omega }\to (0,\mathrm{\infty })}{inf}\left(E_Q\left[\frac{h^{\alpha }}{\alpha }\right]+E_P\left[\frac{h^{\alpha -1}}{1-\alpha }\right]\right),}$$ or, releasing the constraint on ${\displaystyle h}$, $${\displaystyle D_{\alpha }(P\Vert Q)=\frac{1}{\alpha (1-\alpha )}-\underset{h:\mathrm{\Omega }\to \mathbb{R}}{inf}\left(E_Q\left[\frac{|h{|}^{\alpha }}{\alpha }\right]+E_P\left[\frac{|h{|}^{\alpha -1}}{1-\alpha }\right]\right).}$$ Setting ${\displaystyle \alpha =-1}$ yields the variational representation of ${\displaystyle {\chi }^2}$-divergence obtained above.

The domain over which ${\displaystyle h}$ varies is not affine-invariant in general, unlike the ${\displaystyle {\chi }^2}$-divergence case. The ${\displaystyle {\chi }^2}$-divergence is special, since in that case, we can remove the ${\displaystyle |\cdot |}$ from ${\displaystyle |h|}$.

For general ${\displaystyle \alpha \in (-\mathrm{\infty },0)\cup (0,1)}$, the domain over which ${\displaystyle h}$ varies is merely scale invariant. Similar to above, we can replace ${\displaystyle h}$ by ${\displaystyle ah}$, and take minimum over ${\displaystyle a>0}$ to obtain $${\displaystyle D_{\alpha }(P\Vert Q)=\underset{h>0}{sup}\left[\frac{1}{\alpha (1-\alpha )}\left(1-\frac{E_P[h^{\alpha -1}{]}^{\alpha }}{E_Q[h^{\alpha }{]}^{\alpha -1}}\right)\right].}$$ Setting ${\displaystyle \alpha =\frac{1}{2}}$, and performing another substitution by ${\displaystyle g=\sqrt{h}}$, yields two variational representations of the squared Hellinger distance: $${\displaystyle H^2(P\Vert Q)=\frac{1}{2}{D}_{1/2}(P\Vert Q)=2-\underset{h>0}{inf}\left(E_Q\left[h(X)\right]+E_P\left[h(X{)}^{-1}\right]\right),}$$ $${\displaystyle H^2(P\Vert Q)=2\underset{h>0}{sup}\left(1-\sqrt{E_P[h^{-1}]E_Q[h]}\right).}$$ Applying this theorem to the KL-divergence, defined by ${\displaystyle f(x)=x\ln x,f^{\ast }(y)=e^{y-1}}$, yields $${\displaystyle D_{KL}(P;Q)=\underset{g}{sup}{E}_P[g(X)]-e^{-1}{E}_Q[e^{g(X)}].}$$ This is strictly less efficient than the Donsker–Varadhan representation $${\displaystyle D_{KL}(P;Q)=\underset{g}{sup}{E}_P[g(X)]-\ln E_Q[e^{g(X)}].}$$ This defect is fixed by the next theorem.

Improved variational representation

Assume the setup in the beginning of this section ("Variational representations").

Theorem—If ${\displaystyle f(x)=+\mathrm{\infty }}$ on ${\displaystyle x<0}$ (redefine ${\displaystyle f}$ if necessary), then

${\displaystyle D_f(P\Vert Q)=f^{\mathrm{\prime }}(\mathrm{\infty })P\left[S^c\right]+\underset{g}{sup}{\mathbb{E}}_P\left[g{1}_S\right]-{\mathrm{\Psi }}_{Q,P}^{\ast }(g)}$,

where ${\displaystyle {\mathrm{\Psi }}_{Q,P}^{\ast }(g):=\underset{a\in \mathbb{R}}{inf}{\mathbb{E}}_Q\left[f^{\ast }(g(X)-a)\right]+aP[S]}$ and ${\displaystyle S:=\{q>0\}}$, where ${\displaystyle q}$ is the probability density function of ${\displaystyle Q}$ with respect to some underlying measure.

In the special case of ${\displaystyle f^{\mathrm{\prime }}(\mathrm{\infty })=+\mathrm{\infty }}$, we have

${\displaystyle D_f(P\Vert Q)=\underset{g}{sup}{\mathbb{E}}_P[g]-{\mathrm{\Psi }}_Q^{\ast }(g),{\mathrm{\Psi }}_Q^{\ast }(g):=\underset{a\in \mathbb{R}}{inf}{\mathbb{E}}_Q\left[f^{\ast }(g(X)-a)\right]+a}$.

This is Theorem 7.25 in.

Example applications

Applying this theorem to KL-divergence yields the Donsker–Varadhan representation.

Attempting to apply this theorem to the general ${\displaystyle \alpha }$-divergence with ${\displaystyle \alpha \in (-\mathrm{\infty },0)\cup (0,1)}$ does not yield a closed-form solution.

Common examples of f-divergences

The following table lists many of the common divergences between probability distributions and the possible generating functions to which they correspond. Notably, except for total variation distance, all others are special cases of ${\displaystyle \alpha }$-divergence, or linear sums of ${\displaystyle \alpha }$-divergences.

For each f-divergence ${\displaystyle D_f}$, its generating function is not uniquely defined, but only up to ${\displaystyle c\cdot (t-1)}$, where ${\displaystyle c}$ is any real constant. That is, for any ${\displaystyle f}$ that generates an f-divergence, we have ${\displaystyle D_{f(t)}=D_{f(t)+c\cdot (t-1)}}$. This freedom is not only convenient, but actually necessary.

Let ${\displaystyle f_{\alpha }}$ be the generator of ${\displaystyle \alpha }$-divergence, then ${\displaystyle f_{\alpha }}$ and ${\displaystyle f_{1-\alpha }}$ are convex inversions of each other, so ${\displaystyle D_{\alpha }(P\Vert Q)=D_{1-\alpha }(Q\Vert P)}$. In particular, this shows that the squared Hellinger distance and Jensen-Shannon divergence are symmetric.

In the literature, the ${\displaystyle \alpha }$-divergences are sometimes parametrized as

${\displaystyle \{\begin{array}{ll}\frac{4}{1-{\alpha }^2}(1-t^{(1+\alpha )/2}),& \text{if}\alpha \ne \pm 1,\\ t\ln t,& \text{if}\alpha =1,\\ -\ln t,& \text{if}\alpha =-1\end{array}}$

which is equivalent to the parametrization in this page by substituting ${\displaystyle \alpha \leftarrow \frac{\alpha +1}{2}}$.

Relations to other statistical divergences

Here, we compare f-divergences with other statistical divergences.

Rényi divergence

The Rényi divergences is a family of divergences defined by

${\displaystyle R_{\alpha }(P\Vert Q)=\frac{1}{\alpha -1}\log (E_Q\left[{\left(\frac{dP}{dQ}\right)}^{\alpha }\right])}$

when ${\displaystyle \alpha \in (0,1)\cup (1,+\mathrm{\infty })}$. It is extended to the cases of ${\displaystyle \alpha =0,1,+\mathrm{\infty }}$ by taking the limit.

Simple algebra shows that ${\displaystyle R_{\alpha }(P\Vert Q)=\frac{1}{\alpha -1}\ln (1+\alpha (\alpha -1)D_{\alpha }(P\Vert Q))}$, where ${\displaystyle D_{\alpha }}$ is the ${\displaystyle \alpha }$-divergence defined above.

Bregman divergence

The only f-divergence that is also a Bregman divergence is the KL divergence.

Integral probability metrics

The only f-divergence that is also an integral probability metric is the total variation.

Financial interpretation

A pair of probability distributions can be viewed as a game of chance in which one of the distributions defines the official odds and the other contains the actual probabilities. Knowledge of the actual probabilities allows a player to profit from the game. For a large class of rational players the expected profit rate has the same general form as the ƒ-divergence.