corrected some pages

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

Author: JoramSoch

D/beta-data.md

@@ -21,7 +21,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Beta-distributed data are defined as a set of proportions $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ with $y_i \in [0,1], \; i = 1,\ldots,n$, independent and identically distributed according to a [Beta distribution](/D/beta) with shapes $\alpha$ and $\beta$:
+**Definition:** Beta-distributed data are defined as a set of proportions $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ with $y_i \in [0,1], \; i = 1,\ldots,n$, independent and identically distributed according to a [beta distribution](/D/beta) with shapes $\alpha$ and $\beta$:
 
 $$ \label{eq:beta-data}
 y_i \sim \mathrm{Bet}(\alpha,\beta), \quad i = 1, \ldots, n \; .

D/fe.md

@@ -34,7 +34,7 @@ $$ \label{eq:fam}
 f \Leftrightarrow m_1 \vee \ldots \vee m_M \; .
 $$
 
-Then, the family evidence (FE) of $f$ is defined as the [marginal probability](/D/prob-marg) relative to the [model evidences](/D/me) $p(y \vert m_i), conditional only on $f$:
+Then, the family evidence (FE) of $f$ is defined as the [marginal probability](/D/prob-marg) relative to the [model evidences](/D/me) $p(y \vert m_i)$, conditional only on $f$:
 
 $$ \label{eq:fe}
 \mathrm{FE}(f) = p(y|f) \; .

D/lfe.md

@@ -34,13 +34,7 @@ $$ \label{eq:fam}
 f \Leftrightarrow m_1 \vee \ldots \vee m_M \; .
 $$
 
-Then, the family evidence of $f$ is the weighted average of the [model evidences](/D/ml) of $m_1, \ldots, m_M$ where the weights are the within-family [prior model probabilities](/D/prior)
-
-$$ \label{eq:fe}
-p(y|f) = \sum_{i=1}^M p(y|m_i) \, p(m_i|f) \; .
-$$
-
-The log family evidence is given by the logarithm of the family evidence:
+Then, the log family evidence is given by the logarithm of the [family evidence](/D/fe):
 
 $$ \label{eq:lfe}
 \mathrm{LFE}(f) = \log p(y|f) = \log \sum_{i=1}^M p(y|m_i) \, p(m_i|f) \; .

D/lme.md

@@ -28,7 +28,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $m$ be a [full probability model](/D/fpm) with [likelihood function](/D/lf) $p(y \vert \theta, m)$ and [prior distribution](/D/prior) $p(\theta \vert m)$. Then, the log model evidence (LME) of this model is defined as the logarithm of the [marginal likelihood](/D/ml):
+**Definition:** Let $m$ be a [full probability model](/D/fpm) with [likelihood function](/D/lf) $p(y \vert \theta, m)$ and [prior distribution](/D/prior) $p(\theta \vert m)$. Then, the log model evidence (LME) of $m$ is defined as the logarithm of the [marginal likelihood](/D/ml) of this model:
 
 $$ \label{eq:LME}
 \mathrm{LME}(m) = \log p(y|m) \; .

P/beta-chi2.md

@@ -136,7 +136,13 @@ $$ \label{eq:f-Z-s3}
 f_Z(z) = \frac{1}{\mathrm{B}\left( \frac{m}{2}, \frac{n}{2} \right)} \cdot z^{\frac{m}{2}-1} \cdot (1-z)^{\frac{n}{2}-1}
 $$
 
-which is the [probability density function of the beta distribution](/P/beta-pdf), such that
+which is the [probability density function of the beta distribution](/P/beta-pdf) with parameters
+
+$$ \label{eq:beta-chi2-para}
+\alpha = \frac{m}{2} \quad \mathrm{and} \quad \beta = \frac{n}{2} \; ,
+$$
+
+such that
 
 $$ \label{eq:beta-chi2-qed}
 Z \sim \mathrm{Bet}\left( \frac{m}{2}, \frac{n}{2} \right) \; .

P/betabin-mome.md

@@ -58,7 +58,7 @@ m_2 &= \frac{1}{N} \sum_{i=1}^N y_i^2 \; .
 $$
 
 
-**Proof:** The first two [raw moments](/D/mom-raw) of the [beta-binomial distribution](/D/betabin) in terms of the parameters $\alpha$ and $\beta$ are given by
+**Proof:** The first two [raw moments of the beta-binomial distribution](/D/betabin-mom) in terms of the parameters $\alpha$ and $\beta$ are given by
 
 $$ \label{eq:binbeta-mu1-mu2}
 \begin{split}

P/bin-margcond.md

@@ -106,7 +106,7 @@ $$ \label{eq:Y-dist-s7}
 \end{split}
 $$
 
-which is the [probability mass function of the binomial distribution](/P/bin-pmf), such that
+which is the [probability mass function of the binomial distribution](/P/bin-pmf) with parameters $n$ and $pq$, such that
 
 $$ \label{eq:Y-dist-qed}
 Y \sim \mathrm{Bin}(n, pq) \; .

P/lme-der.md

@@ -34,7 +34,7 @@ $$ \label{eq:LME-marg}
 $$
 
 
-**Proof:** This a consequence of the [law of marginal probability](/D/prob-marg) for continuous variables
+**Proof:** This a consequence of the [law of marginal probability](/D/prob-marg) for [continuous variables](/D/rvar-disc)
 
 $$ \label{eq:prob-marg}
 p(y|m) = \int p(y,\theta|m) \, \mathrm{d}\theta