Adjust aspect ratio to 16:9, improve equation numbering, begin consolidating crps slides

index.qmd

@@ -12,6 +12,8 @@ format:
     revealjs:
         embed-resources: false
         footer: ""
+        width: 1280
+        height: 720
         logo: assets/logos_combined.png
         theme: [default, sydney.scss, custom.scss]
         smaller: true
@@ -19,6 +21,17 @@ format:
         slide-number: true
         self-contained-math: true
         crossrefs-hover: true
+        pagetitle: "De-Fence"
+        html-math-method: mathjax
+        include-in-header:
+          - text: |
+              <script>
+              window.MathJax = {
+                tex: {
+                  tags: 'ams'
+                }
+              };
+              </script>   
 execute:
   daemon: false
 highlight-style: github
@@ -342,31 +355,57 @@ Strictly proper for *median* predictions
 ::::
 
 
-## Popular Aggregation Algorithms
+## Popular Algorithms and the Risk
+
+<br/>
+
+:::: {.columns}
+
+::: {.column width="58%"}
+
+### Popular Aggregation Algorithms
+
+<br/>
 
 #### The naive combination
 
-$$
-w_{t,k}^{\text{Naive}} = \frac{1}{K}
-$${#eq-wtk_naive}
+
+\begin{equation}
+w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
+\end{equation}
 
 #### The exponentially weighted average forecaster (EWA)
 
+\begin{equation}
+  \begin{aligned}
+    w_{t,k}^{\text{EWA}} & = \frac{e^{\eta R_{t,k}} }{\sum_{k = 1}^K e^{\eta R_{t,k}}}\\
+    & =
+    \frac{e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k} }{\sum_{k = 1}^K e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k}}
+  \end{aligned}\label{eq:exp_combination}
+\end{equation}
+
+Du kannst dann auch easy darauf verweisen: \ref{eq:exp_combination}.
+
+:::
+
+::: {.column width="2%"}
+
+:::
+
+::: {.column width="38%"}
+
+### Optimality
+
+In stochastic settings, the cumulative Risk should be analyzed `r Citet(my_bib, "wintenberger2017optimal")`:
 \begin{align}
-    w_{t,k}^{\text{EWA}} & = \frac{e^{\eta R_{t,k}} }{\sum_{k = 1}^K e^{\eta R_{t,k}}}
-    =
-    \frac{e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k} }{\sum_{k = 1}^K e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k} }
-    \label{eq_ewa_general}
+    &\underbrace{\widetilde{\mathcal{R}}_t = \sum_{i=1}^t \mathbb{E}[\ell(\widetilde{X}_{i},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Forecaster}} \\
+    &\underbrace{\widehat{\mathcal{R}}_{t,k} = \sum_{i=1}^t \mathbb{E}[\ell(\widehat{X}_{i,k},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Experts}}
+    \label{eq_def_cumrisk}
 \end{align}
 
-#### The polynomial weighted aggregation (PWA)
+:::
 
-\begin{align}
-    w_{t,k}^{\text{PWA}} & = \frac{ 2(R_{t,k})^{q-1}_{+} }{ \|(R_t)_{+}\|^{q-2}_q}
-    \label{eq_pwa_general}
-\end{align}
-
-with $q\geq 2$ and $x_{+}$ the (vector) of positive parts of $x$.
+::::
 
 ## Optimality