Index: The Book of Statistical ProofsGeneral Theorems ▷ Information theory ▷ Kullback-Leibler divergence ▷ Non-symmetry

Theorem: The Kullback-Leibler divergence is non-symmetric, i.e.

$\label{eq:KL-nonsymm} \mathrm{KL}[P||Q] \neq \mathrm{KL}[Q||P]$

for some probability distributions $P$ and $Q$.

Proof: Let $X \in \mathcal{X} = \left\lbrace 0, 1, 2 \right\rbrace$ be a discrete random variable and consider the two probability distributions

$\label{eq:P-Q} \begin{split} P : \, X &\sim \mathrm{Bin}(2, 0.5) \\ Q : \, X &\sim \mathcal{U}(0, 2) \end{split}$

where $\mathrm{Bin}(n, p)$ indicates a binomial distribution and $\mathcal{U}(a, b)$ indicates a discrete uniform distribution.

Then, the probability mass function of the binomial distribution entails that

$\label{eq:p(x)} p(x) = \left\{ \begin{array}{rl} 1/4 \; , & \text{if} \; x = 0 \\ 1/2 \; , & \text{if} \; x = 1 \\ 1/4 \; , & \text{if} \; x = 2 \end{array} \right.$

and the probability mass function of the discrete uniform distribution entails that

$\label{eq:q(x)} q(x) = \frac{1}{3} \; ,$

such that the Kullback-Leibler divergence of $P$ from $Q$ is

$\label{eq:KL-P-Q} \begin{split} \mathrm{KL}[P||Q] &= \sum_{x \in \mathcal{X}} p(x) \cdot \log \frac{p(x)}{q(x)} \\ &= \frac{1}{4} \log \frac{3}{4} + \frac{1}{2} \log \frac{3}{2} + \frac{1}{4} \log \frac{3}{4} \\ &= \frac{1}{2} \log \frac{3}{4} + \frac{1}{2} \log \frac{3}{2} \\ &= \frac{1}{2} \left( \log \frac{3}{4} + \log \frac{3}{2} \right) \\ &= \frac{1}{2} \log \left( \frac{3}{4} \cdot \frac{3}{2} \right) \\ &= \frac{1}{2} \log \frac{9}{8} = 0.0589 \end{split}$

and the Kullback-Leibler divergence of $Q$ from $P$ is

$\label{eq:KL-Q-P} \begin{split} \mathrm{KL}[Q||P] &= \sum_{x \in \mathcal{X}} q(x) \cdot \log \frac{q(x)}{p(x)} \\ &= \frac{1}{3} \log \frac{4}{3} + \frac{1}{3} \log \frac{2}{3} + \frac{1}{3} \log \frac{4}{3} \\ &= \frac{1}{3} \left( \log \frac{4}{3} + \log \frac{2}{3} + \log \frac{4}{3} \right) \\ &= \frac{1}{3} \log \left( \frac{4}{3} \cdot \frac{2}{3} \cdot \frac{4}{3} \right) \\ &= \frac{1}{3} \log \frac{32}{27} = 0.0566 \end{split}$

which provides an example for

$\label{eq:KL-nonsymm-qed} \mathrm{KL}[P||Q] \neq \mathrm{KL}[Q||P]$

and thus proves the theorem.

Sources:

Metadata: ID: P147 | shortcut: kl-nonsymm | author: JoramSoch | date: 2020-08-11, 06:57.