corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

D/logreg.md

@@ -9,7 +9,7 @@ date: 2020-06-28 20:51:00
 
 title: "Logistic regression"
 chapter: "Statistical Models"
-section: "Probability data"
+section: "Categorical data"
 topic: "Logistic regression"
 definition: "Definition"
 

I/Table_of_Contents.md

@@ -391,11 +391,6 @@ title: "Table of Contents"
    &emsp;&ensp; 4.2.1. *[Definition](/D/dir-data)* <br>
    &emsp;&ensp; 4.2.2. **[Maximum likelihood estimation](/P/dir-mle)** <br>
 
-   4.3. Logistic regression <br>
-   &emsp;&ensp; 4.3.1. *[Definition](/D/logreg)* <br>
-   &emsp;&ensp; 4.3.2. **[Probability and log-odds](/P/logreg-pnlo)** <br>
-   &emsp;&ensp; 4.3.3. **[Log-odds and probability](/P/logreg-lonp)** <br>
-
 5. Categorical data
 
    5.1. Binomial observations <br>
@@ -410,6 +405,11 @@ title: "Table of Contents"
    &emsp;&ensp; 5.2.3. **[Posterior distribution](/P/mult-post)** <br>
    &emsp;&ensp; 5.2.4. **[Log model evidence](/P/mult-lme)** <br>
 
+   5.3. Logistic regression <br>
+   &emsp;&ensp; 5.3.1. *[Definition](/D/logreg)* <br>
+   &emsp;&ensp; 5.3.2. **[Probability and log-odds](/P/logreg-pnlo)** <br>
+   &emsp;&ensp; 5.3.3. **[Log-odds and probability](/P/logreg-lonp)** <br>
+
 
 <br>
 <section class="chapter" id="Model Selection">

P/logreg-lonp.md

@@ -9,7 +9,7 @@ date: 2020-03-03 12:01:00
 
 title: "Log-odds and probability in logistic regression"
 chapter: "Statistical Models"
-section: "Probability data"
+section: "Categorical data"
 topic: "Logistic regression"
 theorem: "Log-odds and probability"
 

P/logreg-pnlo.md

@@ -9,7 +9,7 @@ date: 2020-05-19 05:08:00
 
 title: "Probability and log-odds in logistic regression"
 chapter: "Statistical Models"
-section: "Probability data"
+section: "Categorical data"
 topic: "Logistic regression"
 theorem: "Probability and log-odds"
 

P/mean-lotus.md

@@ -53,18 +53,18 @@ $$
 Writing the probability mass function $f_Y(y)$ in terms of $y = g(x)$, we have:
 
 $$ \label{eq:mean-lotus-disc-s2}
-\mathrm{E}[g(X)] = \sum_{y \in \mathcal{Y}} y \, \mathrm{Pr}(g(x) = y) \; .
+\begin{split}
+\mathrm{E}[g(X)] &= \sum_{y \in \mathcal{Y}} y \, \mathrm{Pr}(g(x) = y) \\
+&= \sum_{y \in \mathcal{Y}} y \, \mathrm{Pr}(x = g^{-1}(y)) \\
+&= \sum_{y \in \mathcal{Y}} y \sum_{x = g^{-1}(y)} f_X(x) \\
+&= \sum_{y \in \mathcal{Y}} \sum_{x = g^{-1}(y)} y f_X(x) \\
+&= \sum_{y \in \mathcal{Y}} \sum_{x = g^{-1}(y)} g(x) f_X(x) \; .
+\end{split}
 $$
 
-Replacing the probability that $g(x) = y$ in terms of $f_X(x)$, this becomes:
+Finally, noting that "for all $y$, then for all $x = g^{-1}(y)$" is equivalent to "for all $x$" if $g^{-1}$ is a monotonic function, we can conclude that
 
 $$ \label{eq:mean-lotus-disc-s3}
-\mathrm{E}[g(X)] = \sum_{y \in \mathcal{Y}} y \sum_{x: \; g(x) = y} f_X(x) \; .
-$$
-
-Observe that $y = g(x)$ can be moved into the inner sum. Finally, noting that "for all $y$, then for all $x$, such that $g(x) = y$" is equivalent to "for all $x$" if $g^{-1}$ is a monotonic function, we can conclude that
-
-$$ \label{eq:mean-lotus-disc-s4}
 \mathrm{E}[g(X)] = \sum_{x \in \mathcal{X}} g(x) f_X(x) \; .
 $$
 

P/ng-kl.md

@@ -14,7 +14,7 @@ topic: "Normal-gamma distribution"
 theorem: "Kullback-Leibler divergence"
 
 sources:
-  - authors: "Soch & Allefeld"
+  - authors: "Soch J, Allefeld A"
     year: 2016
     title: "Kullback-Leibler Divergence for the Normal-Gamma Distribution"
     in: "arXiv math.ST"

P/norm-cdfwerf.md

@@ -33,7 +33,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [normal distributions](/D/norm):
+**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [normal distribution](/D/norm):
 
 $$ \label{eq:norm}
 X \sim \mathcal{N}(\mu, \sigma^2) \; .
@@ -104,7 +104,7 @@ T(0) = \varphi(0) \cdot \sum_{i=1}^{\infty} \frac{0^{2i-1}}{(2i-1)!!} + c &= \fr
 \end{split}
 $$
 
-3) Finally, the cumulative distribution functions of the standard normal distribution and the general normal distribution [are related to each other](/P/norm-snorm) as
+3) Finally, the [cumulative distribution functions](/D/cdf) of the [standard normal distribution](/D/snorm) and the general [normal distribution](/D/norm) [are related to each other](/P/norm-snorm) as
 
 $$ \label{eq:norm-snorm-cdf}
 \Phi_{\mu,\sigma}(x) = \Phi\left( \frac{x-\mu}{\sigma} \right) \; .

P/norm-snorm2.md

@@ -62,7 +62,7 @@ where $\mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace$. Wi
 $$ \label{eq:pdf-Z}
 \begin{split}
 f_Z(z) &= \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{g^{-1}(z)-\mu}{\sigma} \right)^2 \right] \cdot \frac{\mathrm{d}g^{-1}(z)}{\mathrm{d}z} \\
-&= \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{(\sigma z + \mu)-\mu}{\sigma} \right)^2 \right] \cdot \frac{\mathrm{d}(\sigma Z + \mu)}{\mathrm{d}z} \\
+&= \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{(\sigma z + \mu)-\mu}{\sigma} \right)^2 \right] \cdot \frac{\mathrm{d}(\sigma z + \mu)}{\mathrm{d}z} \\
 &= \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} z^2 \right] \cdot \sigma \\
 &= \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[ -\frac{1}{2} z^2 \right]
 \end{split}