updates for proper scoring rules and telephone data plot legend fix

slides/advriskmin/rsrc/telephone-data.R

@@ -74,7 +74,7 @@ res.tel$pred_logcosh <- preds.logcosh
 p <- ggplot(res.tel, aes(x = year)) +
   geom_line(aes(y = pred_l2, color = "L2 (OLS)"), size = 1.6, alpha=1) +   
   #geom_line(aes(y = pred_l1, color = "L1"), size = 1.4) +
-  geom_line(aes(y = pred_l1_manual, color = "L1 (man)"), size = 1.6, alpha=1) +
+  geom_line(aes(y = pred_l1_manual, color = "L1"), size = 1.6, alpha=1) +
   geom_line(aes(y = pred_huber, color = "Huber"), size = 1.6, alpha=1) +
   geom_line(aes(y = pred_logcosh, color = "Log-Cosh"), size = 1.6, alpha=1) +
   geom_point(aes(y = calls), color = "black", size = 4, alpha=1) +  

slides/advriskmin/slides-advriskmin-proper-scoring-rules.tex

@@ -105,38 +105,42 @@
 
 \begin{vbframe}{Binary classification scores}
 
-To find strictly proper scores, we can ask: Which functions have the property such that $\E_y[S(\pi,y)]$ is maximized at $\pi=p$? We have
+To find strictly proper scores/losses, we can ask: Which functions have the property such that $\E_y[L(y,\pi)]$ is minimized at $\pi=p$? We have
 
-$$\E_{y}[S(\pi, y)]=p \cdot S(\pi, 1) + (1-p) \cdot S(\pi, 0)$$
+$$\E_{y}[L(y,\pi)]=p \cdot L(1,\pi) + (1-p) \cdot L(0,\pi)$$
 
-Let's further restrict our search to scores $S(\pi,y)$ for which $S(\pi,1)=S(\pi)$ and $S(\pi, 0)=S(1-\pi)$:
+Let's further %restrict our search to scores 
+assume that $L(1,\pi)$ and $L(0, \pi)$ can not be arbitrary, but are the same function evaluated at $\pi$ and $1-\pi$:
+%$S(\pi,y)$ for which 
+$L(1,\pi)=L(\pi)$ and $L(0,\pi)=L(1-\pi)$. Then
 
-$$\E_{y}[S(\pi, y)]=p \cdot S(\pi) + (1-p) \cdot S(1-\pi)$$
+$$\E_{y}[L(y,\pi)]=p \cdot L(\pi) + (1-p) \cdot L(1-\pi)$$
 
 \vspace{0.2cm}
 
-Setting the derivative w.r.t. $\pi$ to $0$ and requiring $\pi=p$ at the optimum, we get the following first-order condition:
+Setting the derivative w.r.t. $\pi$ to $0$ and requiring $\pi=p$ at the optimum (\textbf{propriety}), we get the following first-order condition (F.O.C.):
 
 \vspace{0.3cm}
 
-$$p \cdot S'(p) \overset{!}{=} (1-p) \cdot S'(1-p)$$
+$$p \cdot L'(p) \overset{!}{=} (1-p) \cdot L'(1-p)$$
 
 \framebreak
 
 \begin{itemize}\setlength\itemsep{1.9em}
-    \item F.O.C.:\quad $p \cdot S'(p) \overset{!}{=} (1-p) \cdot S'(1-p)$
-    \item One natural solution is $S'(p)=1/p$, resulting in $p/p=(1-p)/(1-p)=1$ and $S(p)=\log(p)$. 
-    \item This is the \textbf{logarithmic scoring rule} $S(\pi,y)=y \cdot \log(\pi) + (1-y) \cdot \log(1-\pi)$
-    \item Under (loss) minimization this corresponds to the \textbf{log loss} $L(y,\pi)=-(y \cdot \log(\pi) + (1-y) \cdot \log(1-\pi))$
+    \item F.O.C.:\quad $p \cdot L'(p) \overset{!}{=} (1-p) \cdot L'(1-p)$
+    \item One natural solution is $L'(p)=-1/p$, resulting in $-p/p=-(1-p)/(1-p)=-1$ and the antiderivative $L(p)=-\log(p)$. 
+    \item This is the \textbf{log loss} $$L(y,\pi)=-(y \cdot \log(\pi) + (1-y) \cdot \log(1-\pi))$$
+    \item The corresponding scoring rule (maximization) is the strictly proper \textbf{logarithmic scoring rule} $$S(\pi,y)=y \cdot \log(\pi) + (1-y) \cdot \log(1-\pi)$$
 \end{itemize}
 
 \framebreak
 
-\begin{itemize} \setlength\itemsep{1.9em}
-    \item F.O.C.:\quad $p \cdot S'(p) \overset{!}{=} (1-p) \cdot S'(1-p)$
-    \item A second solution is $S'(p)=2(1-p)$, resulting in $2p(1-p)=2(1-p)p$ and $S(p)=-(1-p)^2=-\frac{1}{2}((1-p)^2+(0-(1-p))^2)$
-    \item This gives rise to the \textbf{quadratic scoring rule}, which for two classes is $S(\pi,y)=-\frac{1}{2} \sum_{i=1}^{2}(y_i-\pi_i)^2$\\ {\small (with $y_1=y, y_2=1-y$ and likewise $\pi_1=\pi, \pi_2=1-\pi$)}
-    \item Its positive counterpart is also called the \textbf{Brier score} (minimization) and is effectively the \textbf{MSE loss} for probabilities
+\begin{itemize} \setlength\itemsep{1.2em}
+    \item F.O.C.:\quad $p \cdot L'(p) \overset{!}{=} (1-p) \cdot L'(1-p)$
+    \item A second solution is $L'(p)=-2(1-p)$, resulting in $-2p(1-p)=-2(1-p)p$ and the antiderivative $L(p)=(1-p)^2=\frac{1}{2}((1-p)^2+(0-(1-p))^2)$
+    \item This is also called the \textbf{Brier score} and is effectively the \textbf{MSE loss} for probabilities $$L(y,\pi)=\frac{1}{2}\sum_{i=1}^{2}(y_i-\pi_i)^2$$
+     {\small (with $y_1=y, y_2=1-y$ and likewise $\pi_1=\pi, \pi_2=1-\pi$)}
+    \item Using positive orientation (maximization), this gives rise to the \textbf{quadratic scoring rule}, which for two classes is $S(\pi,y)=-\frac{1}{2} \sum_{i=1}^{2}(y_i-\pi_i)^2$
 \end{itemize}
 
 \end{vbframe}

slides/advriskmin/slides-advriskmin-regression-further-losses.tex

@@ -33,7 +33,7 @@
 
 \begin{vbframe}{Advanced Loss Functions}
 Special loss functions can be used to estimate non-standard posterior components, 
-    to measure errors in a custom way or are designed to have special properties like robustness.
+    to measure errors customarily or which are designed to have special properties like robustness.
 
 \vspace{1cm}
 
@@ -226,7 +226,7 @@
 
 \end{vbframe}
 
-\begin{comment}
+%\begin{comment}
 \begin{vbframe}{Log-Barrier Loss}
 
 \begin{small}
@@ -239,9 +239,9 @@
 \end{small}
 
 \begin{itemize}
-\item Behaves like $L2$ loss for small residuals.
-\item We use this if we don't want residuals larger than $\epsilon$ at all.
-\item No guarantee that the risk minimization problem has a solution.
+\item Behaves like $L2$ loss for small residuals
+\item We use this if we don't want residuals larger than $\epsilon$ at all
+\item No guarantee that the risk minimization problem has a solution
 \item Plot shows log-barrier loss for $\epsilon=2$:
 \end{itemize}
 
@@ -253,38 +253,6 @@
 
 \end{vbframe}
 
-
-
-\begin{vbframe}{Log-Barrier loss}
-
-\begin{itemize}
- % \item Similarly to the Huber loss, there is no closed-form solution for the optimal constant model $f = \thetab$ w.r.t. the log-barrier loss. Numerical optimization is necessary.
- \item Note that the optimization problem has no (finite) solution if there is no way to fit a constant where all residuals are smaller than $\epsilon$.  
-\end{itemize}
-
-\vspace{0.2cm}
-
-\begin{center}
-% \includegraphics[width = 9cm ]{figure_man/log_barrier2.png} 
-\includegraphics[width = \textwidth]{figure/loss_logbarrier_2.png}
-\end{center}
-
-% \framebreak 
-% 
-% 
-% Note that the optimization problem has no (finite) solution if there is no way to fit a constant where all residuals are smaller than $a$. 
-% 
-% \vspace{- 0.2cm}
-% 
-% 
-% \begin{center}
-% \includegraphics[width = 0.8\textwidth]{figure_man/log_barrier_2_1.png} \\
-% \end{center}
-
-% We see that the constant model fitted w.r.t. Huber loss in fact lies between L1- and L2-Loss.
-
-\end{vbframe}
-
 \begin{vbframe}{$\eps$-Insensitive loss}
 
 \vspace*{-0.3cm}
@@ -295,9 +263,9 @@
   \end{cases}, \quad \epsilon \in \R_{+}
 $$
 \begin{itemize}
-\item Modification of $L1$ loss, errors below $\epsilon$ accepted without penalty.
-\item Used in SVM regression.
-\item Properties: convex and not differentiable for $ y - f \in \{-\epsilon, \epsilon\}$.
+\item Modification of $L1$ loss, errors below $\epsilon$ accepted without penalty
+\item Used in SVM regression
+\item Properties: convex and not differentiable for $ y - f \in \{-\epsilon, \epsilon\}$
 \end{itemize}
 
 \vfill
@@ -308,7 +276,7 @@
 \end{center}
 
 \end{vbframe}
-\end{comment}
+%\end{comment}
 
 %% BB: der beweis hier sieht falsch aus...!
 %% \begin{vbframe}{$\epsilon$-insensitive Loss: Optimal Constant}
@@ -354,10 +322,10 @@
 \normalsize
 \begin{itemize}
 \item Extension of $L1$ loss (equal to $L1$ for $\alpha = 0.5$).
-\item Weights either positive or negative residuals more strongly. 
+\item Weighs either positive or negative residuals more strongly
 \item $\alpha<0.5$ $(\alpha>0.5)$ penalty to over-estimation (under-estimation)
 \item Risk minimizer is (conditional) 
-    $\alpha$-quantile (median for $\alpha=0.5$).
+    $\alpha$-quantile (median for $\alpha=0.5$)
 \end{itemize}
 
 \vfill