Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ Continuous uniform distribution ▷ Kullback-Leibler divergence

Theorem: Let $X$ be a random variable. Assume two continuous uniform distributions $P$ and $Q$ specifying the probability distribution of $X$ as

\[\label{eq:cunis} \begin{split} P: \; X &\sim \mathcal{U}(a_1, b_1) \\ Q: \; X &\sim \mathcal{U}(a_2, b_2) \; . \end{split}\]

Then, the Kullback-Leibler divergence of $P$ from $Q$ is given by

\[\label{eq:cuni-KL} \mathrm{KL}[P\,||\,Q] = \ln \frac{b_2-a_2}{b_1-a_1} \; .\]

Proof: The KL divergence for a continuous random variable is given by

\[\label{eq:KL-cont} \mathrm{KL}[P\,||\,Q] = \int_{\mathcal{X}} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x \; .\]

This means that the KL divergence of $P$ from $Q$ is only defined, if for all $x \in \mathcal{X}$, $q(x) = 0$ implies $p(x) = 0$. Thus, $\mathrm{KL}[P\,\vert\vert\,Q]$ only exists, if $a_2 \leq a_1$ and $b_1 \leq b_2$, i.e. if $P$ only places non-zero probability where $Q$ also places non-zero probability, such that $q(x)$ is not zero for any $x \in \mathcal{X}$ where $p(x)$ is positive.

If this requirement is fulfilled, we can write

\[\label{eq:cuni-KL-s1} \mathrm{KL}[P\,||\,Q] = \int_{-\infty}^{a_1} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x + \int_{a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x + \int_{b_1}^{+\infty} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x\]

and because $p(x) = 0$ for any $x < a_1$ and any $x > b_1$, we have

\[\label{eq:cuni-KL-s2} \mathrm{KL}[P\,||\,Q] = \int_{-\infty}^{a_1} 0 \cdot \ln \frac{0}{q(x)} \, \mathrm{d}x + \int_{a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x + \int_{b_1}^{+\infty} 0 \cdot \ln \frac{0}{q(x)} \, \mathrm{d}x \; .\]

Now, $(0 \cdot \ln 0)$ is taken to be zero by convention, such that

\[\label{eq:cuni-KL-s3} \mathrm{KL}[P\,||\,Q] = \int_{a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)} \, \mathrm{d}x\]

and we can use the probability density function of the continuous uniform distribution to evaluate:

\[\label{eq:cuni-KL-s4} \begin{split} \mathrm{KL}[P\,||\,Q] &= \int_{a_1}^{b_1} \frac{1}{b_1-a_1}\, \ln \frac{\frac{1}{b_1-a_1}}{\frac{1}{b_2-a_2}} \, \mathrm{d}x \\ &= \frac{1}{b_1-a_1}\, \ln \frac{b_2-a_2}{b_1-a_1} \int_{a_1}^{b_1} \, \mathrm{d}x \\ &= \frac{1}{b_1-a_1}\, \ln \frac{b_2-a_2}{b_1-a_1} \left[ x \right]_{a_1}^{b_1} \\ &= \frac{1}{b_1-a_1}\, \ln \frac{b_2-a_2}{b_1-a_1} (b_1-a_1) \\ &= \ln \frac{b_2-a_2}{b_1-a_1} \; . \end{split}\]

Metadata: ID: P422 | shortcut: cuni-kl | author: JoramSoch | date: 2023-10-27, 12:32.