update exercises week 38

doc/LectureNotes/_build/html/_sources/exercisesweek38.ipynb

@@ -178,20 +178,12 @@
     "We see that $\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\mathbb{E} \\big[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}\\big ]$ for any $\\lambda > 0$.\n"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "id": "65f6f914",
-   "metadata": {},
-   "source": [
-    "**b)** Why do we say that Ridge regression gives a biased estimate? Is this a problem?\n"
-   ]
-  },
   {
    "cell_type": "markdown",
    "id": "b4e721fc",
    "metadata": {},
    "source": [
-    "**c)** Show that the variance is\n"
+    "**b)** Show that the variance is\n"
    ]
   },
   {
@@ -297,7 +289,8 @@
    "source": [
     "$$\n",
     "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
-    "$$\n"
+    "$$\n",
+    "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$."
    ]
   },
   {

doc/LectureNotes/_build/html/exercisesweek38.html

@@ -480,8 +480,7 @@ A given parameter <span class="math notranslate nohighlight">\({\boldsymbol{\hat
 \mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}
 \]</div>
 <p>We see that <span class="math notranslate nohighlight">\(\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big] \not= \mathbb{E} \big[\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}\big ]\)</span> for any <span class="math notranslate nohighlight">\(\lambda &gt; 0\)</span>.</p>
-<p><strong>b)</strong> Why do we say that Ridge regression gives a biased estimate? Is this a problem?</p>
-<p><strong>c)</strong> Show that the variance is</p>
+<p><strong>b)</strong> Show that the variance is</p>
 <div class="math notranslate nohighlight">
 \[
 \mathbf{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T}\mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}
@@ -517,6 +516,7 @@ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_
 \[
 \mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
 \]</div>
+<p>In order to arrive at the equation for the bias, we have to approximate the unknown function <span class="math notranslate nohighlight">\(f\)</span> with the output/target values <span class="math notranslate nohighlight">\(y\)</span>.</p>
 <p><strong>b)</strong> Explain what the terms mean and discuss their interpretations.</p>
 </section>
 <section id="exercise-4-computing-the-bias-and-variance">

doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb

@@ -178,20 +178,12 @@
     "We see that $\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\mathbb{E} \\big[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}\\big ]$ for any $\\lambda > 0$.\n"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "id": "65f6f914",
-   "metadata": {},
-   "source": [
-    "**b)** Why do we say that Ridge regression gives a biased estimate? Is this a problem?\n"
-   ]
-  },
   {
    "cell_type": "markdown",
    "id": "b4e721fc",
    "metadata": {},
    "source": [
-    "**c)** Show that the variance is\n"
+    "**b)** Show that the variance is\n"
    ]
   },
   {
@@ -297,7 +289,8 @@
    "source": [
     "$$\n",
     "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
-    "$$\n"
+    "$$\n",
+    "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$."
    ]
   },
   {

doc/LectureNotes/exercisesweek38.ipynb

@@ -290,7 +290,7 @@
     "$$\n",
     "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
     "$$\n",
-    "In order to arrive at the last equation, we have to approximate the unknown function $f$ with the output/target values $y$."
+    "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$."
    ]
   },
   {