added proof "duni-kl"

Author: JoramSoch

P/duni-kl.md

@@ -0,0 +1,75 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2023-11-17 15:19:37
+
+title: "Kullback-Leibler divergence for the discrete uniform distribution"
+chapter: "Probability Distributions"
+section: "Univariate discrete distributions"
+topic: "Discrete uniform distribution"
+theorem: "Kullback-Leibler divergence"
+
+sources:
+
+proof_id: "P425"
+shortcut: "duni-kl"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a [random variable](/D/rvar). Assume two [discrete uniform distributions](/D/Duni) $P$ and $Q$ specifying the probability distribution of $X$ as
+
+$$ \label{eq:dunis}
+\begin{split}
+P: \; X &\sim \mathcal{U}(a_1, b_1) \\
+Q: \; X &\sim \mathcal{U}(a_2, b_2) \; .
+\end{split}
+$$
+
+Then, the [Kullback-Leibler divergence](/D/kl) of $P$ from $Q$ is given by
+
+$$ \label{eq:duni-KL}
+\mathrm{KL}[P\,||\,Q] = \ln \frac{b_2-a_2+1}{b_1-a_1+1} \; .
+$$
+
+
+**Proof:** The [KL divergence for a discrete random variable](/D/kl) is given by
+
+$$ \label{eq:KL-disc}
+\mathrm{KL}[P\,||\,Q] = \sum_{x \in \mathcal{X}} p(x) \, \ln \frac{p(x)}{q(x)} \; .
+$$
+
+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, $\mathrm{KL}[P\,||\,Q]$ 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
+
+$$ \label{eq:duni-KL-s1}
+\mathrm{KL}[P\,||\,Q] = \sum_{x=-\infty}^{a_1} p(x) \, \ln \frac{p(x)}{q(x)} + \sum_{x=a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)} + \sum_{x=b_1}^{+\infty} p(x) \, \ln \frac{p(x)}{q(x)}
+$$
+
+and because $p(x) = 0$ for any $x < a_1$ and any $x > b_1$, we have
+
+$$ \label{eq:duni-KL-s2}
+\mathrm{KL}[P\,||\,Q] = \sum_{x=-\infty}^{a_1} 0 \cdot \ln \frac{0}{q(x)} + \sum_{x=a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)} + \sum_{x=b_1}^{+\infty} 0 \cdot \ln \frac{0}{q(x)} \; .
+$$
+
+Now, $(0 \cdot \ln 0)$ is taken to be $0$ [by convention](/D/ent), such that
+
+$$ \label{eq:duni-KL-s3}
+\mathrm{KL}[P\,||\,Q] = \sum_{x=a_1}^{b_1} p(x) \, \ln \frac{p(x)}{q(x)}
+$$
+
+and we can use the [probability mass function of the discrete uniform distribution](/P/duni-pmf) to evaluate:
+
+$$ \label{eq:duni-KL-s4}
+\begin{split}
+\mathrm{KL}[P\,||\,Q] &= \sum_{x=a_1}^{b_1} \frac{1}{b_1-a_1+1} \cdot \ln \frac{\frac{1}{b_1-a_1+1}}{\frac{1}{b_2-a_2+1}} \\
+&= \frac{1}{b_1-a_1+1} \cdot \ln \frac{b_2-a_2+1}{b_1-a_1+1} \sum_{x=a_1}^{b_1} 1 \\
+&= \frac{1}{b_1-a_1+1} \cdot \ln \frac{b_2-a_2+1}{b_1-a_1+1} \cdot (b_1-a_1+1) \\
+&= \ln \frac{b_2-a_2+1}{b_1-a_1+1} \; .
+\end{split}
+$$