corrected some pages

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

Author: JoramSoch

P/bin-margcond.md

@@ -64,7 +64,7 @@ $$ \label{eq:Y-dist-s3}
 \mathrm{Pr}(Y = m) = \sum_{k=m}^{n} {k \choose m} \, q^m \, (1-q)^{k-m} \cdot {n \choose k} \, p^k \, (1-p)^{n-k} \; .
 $$
 
-Applying the binomial coefficient identity ${n \choose k} {k \choose m} = {n \choose m} {n-m \choose k-m}$ rearranging the terms, we have:
+Applying the binomial coefficient identity ${n \choose k} {k \choose m} = {n \choose m} {n-m \choose k-m}$ and rearranging the terms, we have:
 
 $$ \label{eq:Y-dist-s4}
 \mathrm{Pr}(Y = m) = \sum_{k=m}^{n} {n \choose m} \, {n-m \choose k-m} \, p^k \, q^m \, (1-p)^{n-k} \, (1-q)^{k-m} \; .

P/lbf-lme.md

@@ -43,19 +43,23 @@ $$
 
 and the [log Bayes factor](/D/lbf) is defined as the logarithm of the Bayes factor
 
-$$ \label{eq:LBF}
-\mathrm{LBF}_{12} = \log \mathrm{BF}_{12} = \log \frac{p(y|m_1)}{p(y|m_2)} \; .
+$$ \label{eq:LBF-s1}
+\mathrm{LBF}_{12} = \log \mathrm{BF}_{12} \; .
 $$
 
-With the definition of the [log model evidence](/D/lme)
+Thus, the log Bayes factor can be expressed as
+
+$$ \label{eq:LBF-s2}
+\mathrm{LBF}_{12} = \log \frac{p(y|m_1)}{p(y|m_2)} \; .
+$$
+
+and, with the definition of the [log model evidence](/D/lme)
 
 $$ \label{eq:LME}
 \mathrm{LME}(m) = \log p(y|m)
 $$
 
-the log Bayes factor can be expressed as:
-
-Resolving the logarithm and applying the definition of the [log model evidence](/D/lme), we finally have:
+and resolving the logarithm, we finally have:
 
 $$ \label{eq:LBF-LME-qed}
 \begin{split}

P/matn-marg.md

@@ -163,7 +163,7 @@ x_{ij} &= AXB = e_i X e_j^\mathrm{T} \\
 \end{split}
 $$
 
-As $x_{ij}$ is a scalar, this is equivalent to [a univariate normal distribution](/D/norm) as [a special case of](/P/norm-mvn) of [the matrix-normal distribution](/P/mvn-matn):
+As $x_{ij}$ is a scalar, this is equivalent to [a univariate normal distribution](/D/norm) as [a special case](/P/norm-mvn) of [the matrix-normal distribution](/P/mvn-matn):
 
 $$ \label{eq:xij-marg-norm}
 x_{ij} \sim \mathcal{N}(m_{ij}, u_{ii} v_{jj}) \; .

P/norm-chi2.md

@@ -76,8 +76,8 @@ $$ \label{eq:sum-Ui2-s1}
 \begin{split}
 \sum_{i=1}^{n} U_i^2 &= \sum_{i=1}^{n} \left( \frac{X_i - \mu}{\sigma} \right)^2 \\
 &= \sum_{i=1}^{n} \left( \frac{(X_i - \bar{X}) + (\bar{X} - \mu)}{\sigma} \right)^2 \\
-&= \sum_{i=1}^{n} \frac{(X_i - \bar{X})^2}{\sigma^2} + \sum_{i=1}^{n} \frac{(\bar{X} - \mu)^2}{\sigma^2} + \sum_{i=1}^{n} \frac{(X_i - \bar{X})(\bar{X} - \mu)}{\sigma^2} \\
-&= \sum_{i=1}^{n} \left( \frac{X_i - \bar{X}}{\sigma^2} \right)^2 + \sum_{i=1}^{n} \left( \frac{\bar{X} - \mu}{\sigma^2} \right)^2 + \frac{(\bar{X} - \mu)}{\sigma^2} \sum_{i=1}^{n} (X_i - \bar{X}) \; .
+&= \sum_{i=1}^{n} \frac{(X_i - \bar{X})^2}{\sigma^2} + \sum_{i=1}^{n} \frac{(\bar{X} - \mu)^2}{\sigma^2} + 2 \sum_{i=1}^{n} \frac{(X_i - \bar{X})(\bar{X} - \mu)}{\sigma^2} \\
+&= \sum_{i=1}^{n} \left( \frac{X_i - \bar{X}}{\sigma} \right)^2 + \sum_{i=1}^{n} \left( \frac{\bar{X} - \mu}{\sigma} \right)^2 + \frac{(\bar{X} - \mu)}{\sigma^2} \sum_{i=1}^{n} (X_i - \bar{X}) \; .
 \end{split}
 $$