# Relative Entropy

## Alexei Gilchrist

The relative entropy, or Kullback-Leibler divergence is a measure of the difference of two distributions

### 1 Definition

The relative entropy of two distributions $$p(x)$$ and $$q(x)$$ is defined as

$\begin{equation*}D(p||q) = \sum_{x} p(x) \log \frac{p(x)}{q(x)}\end{equation*}$
It’s also known as the Kullback-Leibler divergence or distance.

Note that it’s not a true metric or distance as it’s not symmetric for starters.

### 2 `Derivation’

The following is a plausible story that would lead you to the relative entropy. Say the true probabilities in some problem are

$\begin{equation*}\{p(x_{1}), p(x_{2}), \ldots p(x_{n})\}\end{equation*}$
but we believe we have
$\begin{equation*}\{q(x_{1}), q(x_{2}), \ldots q(x_{n})\}\end{equation*}$
so that with proposition $$x$$ we mistakenly associate the information
$\begin{equation*}I_{q}(x) = -\log q(x).\end{equation*}$
The erroneous average information will be
$\begin{equation*}\langle I_{q}(x) \rangle_{x} = -\langle \log(q(x)) \rangle_{x} = -\sum p(x)\log(q(x))\end{equation*}$
since the symbols are still occurring with probabilities $$p(x)$$.

The real average information is of course the entropy of $$p(x)$$:

$\begin{equation*}\langle I_{p}(x) \rangle_{x} = -\sum p(x)\log(p(x)) = H(x).\end{equation*}$
The discrepancy between these two is the relative entropy:
\begin{align*}I_{q}(x)-I_{p}(x) &= \left\langle \log\frac{1}{q(x)} \right\rangle_{x} - \left\langle \log\frac{1}{p(x)} \right\rangle_{x} \\ &= -\sum p(x)\log(q(x)) +\sum p(x)\log(p(x))\\ &= \sum_{x} p(x) \log \frac{p(x)}{q(x)} \\ &= D(p||q)\end{align*}

### 3 Properties

1. Not symmetric: $$D(p||q)\ne D(q||p)$$. Naively you might have thought that mistaking $$p$$ for $$q$$ was the same as mistaking $$q$$ for $$p$$ but this is not the case. Think about the consequences of mistaking a flat distribution with a sharply peaked one and vice versa.
2. $$D(p||q)\ge 0$$ with equality implying $$p(x)=q(x)$$ for all $$x$$.

The proof for the last item follows from Jensen’s inequality.

Proof

First start by writing the relative entropy as

$\begin{equation*}D(p||q) = -\sum_{x} p(x) \log \frac{q(x)}{p(x)} = \sum_{x} p(x) f(u)\end{equation*}$
where we have inverted the fraction in the logarithm an brought out an overall minus sign. The function $$f(x)=-\log(u_{x})$$ is a strictly convex function of $$u_{x}=q(x)/p(x)$$ and the last term is the average of $$f$$ so we can apply Jensen’s inequality,
\begin{align*}D(p||q) &\ge f\left(\sum_{x}p(x)u_{x}\right)\\ &= -\log\left(\sum_{x}p(x)\frac{q(x)}{p(x)}\right) \\ &= -\log(1) = 0.\end{align*}
Lastly, when $$D(p||q) =0$$, since $$f$$ is strictly convex we have $$u_{x}=\langle u_{x}\rangle$$, so $$q(x)/p(x)=1$$ or $$q(x)=p(x)$$.