Merge overleaf-2024-11-07-0946 into master

slides/supervised-classification/slides-classification-discranalysis.tex

@@ -45,24 +45,23 @@
 \begin{vbframe}{Univariate example}
 \begin{small}
 \begin{itemize}
-\item Assume we observed the body height in cm of 10 men and 10 women. 
-\item For a given body height of a new person, we want to classify that person to the classes \textit{man} or \textit{woman}. 
-\item LDA fits a normal distribution for both classes with equal standard deviations, i.e., both distributions have the same shape.
+\item Given the heights (in cm) of 10 men and 10 women, we aim to classify a new person as male or female based on their height.
+\item LDA models both classes using normal distributions with equal standard deviations (identical shapes).
 \end{itemize}
 \begin{center}
 \includegraphics[width=0.85\textwidth, clip=true, trim={0 0 0 0}]{figure/disc_univariate-1.png}
 \end{center}
-The optimal separation is located at the intersection (= decision boundary)!
+\centerline{The optimal separation is located at the intersection (= decision boundary)!}
 \end{small}
 \end{vbframe}
 
 \begin{vbframe}{Univariate example: equal class sizes}
 \begin{small}
-We can use our learned distributions to answer questions, e.g., how likely is somebody with body height of 172 cm a male person?
+Using our learned distributions, we can compute the posterior probability that a 172 cm tall person is male.
 \begin{center}
 \includegraphics[width=0.85\textwidth, clip=true, trim={0 0 0 0}]{figure/disc_univariate-2.png}
 \end{center}
-For equal class sizes, the prior probs $\pik$ cancel out:
+For equal class sizes, the prior probs $\pik$ cancel out (since $\pi_{man} = \pi_{woman}$):
 $$
 \P(y = \text{man} \mid \xv) = \frac{p(\xv \mid y = \text{man})}{p(\xv \mid y = \text{man}) + p(\xv \mid y = \text{woman})} = \frac{0.0135}{0.0135 + 0.088} = 0.133
 $$
@@ -71,14 +70,17 @@
 
 \begin{vbframe}{Univariate example: unequal class sizes}
 \begin{small}
-\begin{itemize}
-  \item Usually class sizes are not always equal, e.g., a class might have a higher occurence.
-  \item The formula to compute the \textit{posterior probability} does not simplify and the optimal linear separation will be shifted to the class with lower occurence.
-\end{itemize}
+For unequal class sizes (e.g., $\pi_{woman} = 2\pi_{man}$), the prior probs matter and cause a shift of the decision boundary towards the smaller class.
 \begin{center}
-\includegraphics[width=\textwidth, clip=true, trim={0 0 0 0}]{figure/disc_univariate-3.png}
+\includegraphics[width=0.86\textwidth, clip=true, trim={0 0 0 0}]{figure/disc_univariate-3.png}
 \end{center}
+\begin{align*}
+\P(y = \text{man} \mid \xv) &= \frac{p(\xv \mid y = \text{man}) \pi_{man}}{p(\xv \mid y = \text{man}) \pi_{man} + p(\xv \mid y = \text{woman}) \pi_{woman}}\\
+&=\frac{0.0135 \tfrac{1}{3}}{0.0135 \tfrac{1}{3} + 0.088 \tfrac{2}{3}} = 0.0712
+\end{align*}
+
 \end{small}
+
 \end{vbframe}
 \begin{vbframe}{LDA decision boundaries}
 
@@ -146,7 +148,7 @@
 
 \end{vbframe}
 
-\begin{vbframe}{QDA in 1D}
+\begin{vbframe}{Univariate Example with QDA}
 \begin{small}
 Different covariance matrices lead to multiple classification rules:
 \begin{itemize}