diff --git a/doc/pub/week37/html/._week37-bs024.html b/doc/pub/week37/html/._week37-bs024.html index e9fdfa5c1..05b66397b 100644 --- a/doc/pub/week37/html/._week37-bs024.html +++ b/doc/pub/week37/html/._week37-bs024.html @@ -287,12 +287,12 @@ MathJax.Hub.Config({

Suppose we have a PDF \( p(x) \) from which we generate a series \( N \) -of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \) +of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \) is viewed as the average of a specific measurement, e.g., throwing dice 100 times and then taking the average value, or producing a certain amount of random numbers. -For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion -which follows. +For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion +which follows. We do the same for \( \mathbb{E}[z]=z \).

If we compute the mean \( z \) of \( m \) such mean values \( x_i \) diff --git a/doc/pub/week37/html/._week37-bs028.html b/doc/pub/week37/html/._week37-bs028.html index 0794526fc..1e4230c5d 100644 --- a/doc/pub/week37/html/._week37-bs028.html +++ b/doc/pub/week37/html/._week37-bs028.html @@ -304,7 +304,7 @@ $$

The latter is true only if the average value is known exactly. This is obtained in the limit \( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain -the familiar expression in statistics +the familiar expression in statistics (the so-called Bessel correction) $$ \sigma_m\approx \frac{\sigma}{\sqrt{m-1}}. diff --git a/doc/pub/week37/html/._week37-bs037.html b/doc/pub/week37/html/._week37-bs037.html index 39584b03e..0e7980942 100644 --- a/doc/pub/week37/html/._week37-bs037.html +++ b/doc/pub/week37/html/._week37-bs037.html @@ -289,8 +289,8 @@ MathJax.Hub.Config({ We will discuss the bias-variance tradeoff in the context of continuous predictions such as regression. However, many of the intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset \( \mathcal{L} \) consisting of the data -\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). +tasks. Consider a dataset \( \mathcal{D} \) consisting of the data +\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).

Let us assume that the true data is generated from a noisy model diff --git a/doc/pub/week37/html/._week37-bs044.html b/doc/pub/week37/html/._week37-bs044.html index a75a607e8..6a6cb420e 100644 --- a/doc/pub/week37/html/._week37-bs044.html +++ b/doc/pub/week37/html/._week37-bs044.html @@ -301,7 +301,7 @@ $$

diff --git a/doc/pub/week37/html/week37-reveal.html b/doc/pub/week37/html/week37-reveal.html index 3000dce08..1ac93f531 100644 --- a/doc/pub/week37/html/week37-reveal.html +++ b/doc/pub/week37/html/week37-reveal.html @@ -906,12 +906,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i

Suppose we have a PDF \( p(x) \) from which we generate a series \( N \) -of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \) +of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \) is viewed as the average of a specific measurement, e.g., throwing dice 100 times and then taking the average value, or producing a certain amount of random numbers. -For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion -which follows. +For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion +which follows. We do the same for \( \mathbb{E}[z]=z \).

If we compute the mean \( z \) of \( m \) such mean values \( x_i \) @@ -1044,7 +1044,7 @@ $$

The latter is true only if the average value is known exactly. This is obtained in the limit \( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain -the familiar expression in statistics +the familiar expression in statistics (the so-called Bessel correction)

 
$$ \sigma_m\approx @@ -1282,8 +1282,8 @@ plt.show() We will discuss the bias-variance tradeoff in the context of continuous predictions such as regression. However, many of the intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset \( \mathcal{L} \) consisting of the data -\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). +tasks. Consider a dataset \( \mathcal{D} \) consisting of the data +\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).

Let us assume that the true data is generated from a noisy model @@ -1637,7 +1637,7 @@ $$

diff --git a/doc/pub/week37/html/week37-solarized.html b/doc/pub/week37/html/week37-solarized.html index 92297bafd..da192c64a 100644 --- a/doc/pub/week37/html/week37-solarized.html +++ b/doc/pub/week37/html/week37-solarized.html @@ -949,12 +949,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i

Suppose we have a PDF \( p(x) \) from which we generate a series \( N \) -of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \) +of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \) is viewed as the average of a specific measurement, e.g., throwing dice 100 times and then taking the average value, or producing a certain amount of random numbers. -For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion -which follows. +For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion +which follows. We do the same for \( \mathbb{E}[z]=z \).

If we compute the mean \( z \) of \( m \) such mean values \( x_i \) @@ -1068,7 +1068,7 @@ $$

The latter is true only if the average value is known exactly. This is obtained in the limit \( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain -the familiar expression in statistics +the familiar expression in statistics (the so-called Bessel correction) $$ \sigma_m\approx \frac{\sigma}{\sqrt{m-1}}. @@ -1299,8 +1299,8 @@ plt.show() We will discuss the bias-variance tradeoff in the context of continuous predictions such as regression. However, many of the intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset \( \mathcal{L} \) consisting of the data -\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). +tasks. Consider a dataset \( \mathcal{D} \) consisting of the data +\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).

Let us assume that the true data is generated from a noisy model @@ -1638,7 +1638,7 @@ $$

diff --git a/doc/pub/week37/html/week37.html b/doc/pub/week37/html/week37.html index 5e7a20f6f..81e0cf3e7 100644 --- a/doc/pub/week37/html/week37.html +++ b/doc/pub/week37/html/week37.html @@ -954,12 +954,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i

Suppose we have a PDF \( p(x) \) from which we generate a series \( N \) -of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \) +of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \) is viewed as the average of a specific measurement, e.g., throwing dice 100 times and then taking the average value, or producing a certain amount of random numbers. -For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion -which follows. +For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion +which follows. We do the same for \( \mathbb{E}[z]=z \).

If we compute the mean \( z \) of \( m \) such mean values \( x_i \) @@ -1073,7 +1073,7 @@ $$

The latter is true only if the average value is known exactly. This is obtained in the limit \( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain -the familiar expression in statistics +the familiar expression in statistics (the so-called Bessel correction) $$ \sigma_m\approx \frac{\sigma}{\sqrt{m-1}}. @@ -1304,8 +1304,8 @@ plt.show() We will discuss the bias-variance tradeoff in the context of continuous predictions such as regression. However, many of the intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset \( \mathcal{L} \) consisting of the data -\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \). +tasks. Consider a dataset \( \mathcal{D} \) consisting of the data +\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).

Let us assume that the true data is generated from a noisy model @@ -1643,7 +1643,7 @@ $$

diff --git a/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz b/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz index 9a14287c6..71381324f 100644 Binary files a/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz and b/doc/pub/week37/ipynb/ipynb-week37-src.tar.gz differ diff --git a/doc/pub/week37/ipynb/week37.ipynb b/doc/pub/week37/ipynb/week37.ipynb index ea7af2c3b..cf68b2627 100644 --- a/doc/pub/week37/ipynb/week37.ipynb +++ b/doc/pub/week37/ipynb/week37.ipynb @@ -890,12 +890,12 @@ "\n", "\n", "Suppose we have a PDF $p(x)$ from which we generate a series $N$\n", - "of averages $\\langle x_i \\rangle$. Each mean value $\\langle x_i \\rangle$\n", + "of averages $\\mathbb{E}[x_i]$. Each mean value $\\mathbb{E}[x_i]$\n", "is viewed as the average of a specific measurement, e.g., throwing \n", "dice 100 times and then taking the average value, or producing a certain\n", "amount of random numbers. \n", - "For notational ease, we set $\\langle x_i \\rangle=x_i$ in the discussion\n", - "which follows. \n", + "For notational ease, we set $\\mathbb{E}[x_i]=x_i$ in the discussion\n", + "which follows. We do the same for $\\mathbb{E}[z]=z$.\n", "\n", "If we compute the mean $z$ of $m$ such mean values $x_i$" ] @@ -1085,7 +1085,7 @@ "source": [ "The latter is true only if the average value is known exactly. This is obtained in the limit\n", "$m\\rightarrow \\infty$ only. Because the mean and the variance are measured quantities we obtain \n", - "the familiar expression in statistics" + "the familiar expression in statistics (the so-called Bessel correction)" ] }, { @@ -1325,8 +1325,8 @@ "We will discuss the bias-variance tradeoff in the context of\n", "continuous predictions such as regression. However, many of the\n", "intuitions and ideas discussed here also carry over to classification\n", - "tasks. Consider a dataset $\\mathcal{L}$ consisting of the data\n", - "$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$. \n", + "tasks. Consider a dataset $\\mathcal{D}$ consisting of the data\n", + "$\\mathbf{X}_\\mathcal{D}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$. \n", "\n", "Let us assume that the true data is generated from a noisy model" ] @@ -1757,7 +1757,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", + "* Evaluate the prediction performance of these models on the test set by $[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", "\n", "* Repeat the first three steps such that each sample plays the role of the test set once.\n", "\n", diff --git a/doc/src/week37/week37.do.txt b/doc/src/week37/week37.do.txt index e3b9e3bd1..c59ccdbc1 100644 --- a/doc/src/week37/week37.do.txt +++ b/doc/src/week37/week37.do.txt @@ -643,12 +643,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i Suppose we have a PDF $p(x)$ from which we generate a series $N$ -of averages $\langle x_i \rangle$. Each mean value $\langle x_i \rangle$ +of averages $\mathbb{E}[x_i]$. Each mean value $\mathbb{E}[x_i]$ is viewed as the average of a specific measurement, e.g., throwing dice 100 times and then taking the average value, or producing a certain amount of random numbers. -For notational ease, we set $\langle x_i \rangle=x_i$ in the discussion -which follows. +For notational ease, we set $\mathbb{E}[x_i]=x_i$ in the discussion +which follows. We do the same for $\mathbb{E}[z]=z$. If we compute the mean $z$ of $m$ such mean values $x_i$ !bt @@ -759,7 +759,7 @@ standard deviation, given by The latter is true only if the average value is known exactly. This is obtained in the limit $m\rightarrow \infty$ only. Because the mean and the variance are measured quantities we obtain -the familiar expression in statistics +the familiar expression in statistics (the so-called Bessel correction) !bt \[ \sigma_m\approx @@ -962,8 +962,8 @@ plt.show() We will discuss the bias-variance tradeoff in the context of continuous predictions such as regression. However, many of the intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset $\mathcal{L}$ consisting of the data -$\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}$. +tasks. Consider a dataset $\mathcal{D}$ consisting of the data +$\mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}$. Let us assume that the true data is generated from a noisy model @@ -1295,7 +1295,7 @@ cross-validation (LOOCV). \end{align*} !et -* Evaluate the prediction performance of these models on the test set by $\log\{L[y_i, \bm{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]\}$. Or, by the prediction error $|y_i - \bm{X}_{i, \ast} \bm{\beta}_{-i}(\lambda)|$, the relative error, the error squared or the R2 score function. +* Evaluate the prediction performance of these models on the test set by $[y_i, \bm{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]$. Or, by the prediction error $|y_i - \bm{X}_{i, \ast} \bm{\beta}_{-i}(\lambda)|$, the relative error, the error squared or the R2 score function. * Repeat the first three steps such that each sample plays the role of the test set once.