some logarithm and conditional mutual entropy

Author: jhobbs

content/algebra/logarithms.tex

@@ -10,7 +10,9 @@
 \section{Logarithms}
 
 \begin{definition}[Logarithm]
-	For $b > 1,$ $y > 0$ the unique \@{real} $x$ such that $b^x = y$ is called the \textbf{logarithm of} $y$ \textbf{to the base} $b$.
+	For $b > 1,$ $y > 0$ the unique \@{real} $x$ such that $b^x = y$ is called the \textbf{logarithm of} $y$ \textbf{to the base} $b$ and is denoted as
+
+\[ \log_b{(y)}. \]
 \end{definition}
 
 \begin{note}
@@ -33,6 +35,11 @@ \section{Logarithms}
 \[ \log_b{(a)} = \frac{\log_x{(a)}}{\log_x{(b)}}. \]
 \end{theorem}
 
+\begin{proof}
+By definition,
+\end{proof}
+
+
 \begin{theorem}[Log is Concave]\label{log-is-concave}
 The $\log$ \@{function} is \@[concave]{concave-function}, i.e. $-\log$ is \@[convex]{convex-function}.
 \end{theorem}

content/applied-math/information-theory/01-discrete-entropy.tex

@@ -481,7 +481,15 @@ \subsection{Chain Rules}
 \end{theorem}
 
 \begin{definition}[Conditional Mutual Information]
-TODO: Cover \& Thomas, Section 2.5.
+	The \textbf{conditional mutual information} of \@{random-variables} $X$ and $Y$ given $Z$ is defined by
+
+\bal
+	I(X;Y  | Z) & = H(X|Z) - H(X|Y, Z) \\
+	             & = E_{p(x,y,z)} \log{ \frac{p(X, Y|Z)}{p(X|Z)p(Y|Z)} }.
+\eal
+
+It is the \@{uncertainty} about $X$ after learning $Z$ minus the uncertainty about $X$ after learning $Y$ and $Z$.
+
 \end{definition}
 
 \begin{theorem}[Chain Rule for Mutual Information]