corrected some pages

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

Author: JoramSoch

P/bin-lbf.md

@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
 date: 2022-11-25 14:40:00
 
-title: "Log model evidence for binomial observations"
+title: "Log Bayes factor for binomial observations"
 chapter: "Statistical Models"
 section: "Count data"
 topic: "Binomial observations"

P/bin-lme.md

@@ -118,7 +118,7 @@ the LME finally becomes
 
 $$ \label{eq:Bin-LME-s2}
 \begin{split}
-\log \mathrm{p}(y|m) = \log \Gamma(n+1) &- \log \Gamma(k+1) - \log \Gamma(n-k+1) \\
+\log \mathrm{p}(y|m) = \log \Gamma(n+1) &- \log \Gamma(y+1) - \log \Gamma(n-y+1) \\
 &+ \log B(\alpha_n,\beta_n) - \log B(\alpha_0,\beta_0) \; .
 \end{split}
 $$

P/bin-mll.md

@@ -39,8 +39,6 @@ $$
 
 **Proof:** The [log-likelihood function for binomial data](/P/bin-mle) is given by
 
-With the [probability mass function of the binomial distribution](/P/bin-pmf), equation \eqref{eq:Bin} implies the following [likelihood function](/D/lf):
-
 $$ \label{eq:Bin-LL}
 \mathrm{LL}(p) = \log {n \choose y} + y \log p + (n-y) \log (1-p)
 $$

P/mult-lbf.md

@@ -64,7 +64,7 @@ $$ \label{eq:Mult-LME-m1}
 \end{split}
 $$
 
-Because the null model $m_0$ has no free parameter, its [log model evidence](/D/lme) (logarithmized [marginal likelihood](/D/ml)) is equal to the [log-likelihood function for multinomial observations](/P/mult-mle) at the value $p = [1/k, \ldots, 1/k]$:
+Because the null model $m_0$ has no free parameter, its [log model evidence](/D/lme) (logarithmized [marginal likelihood](/D/ml)) is equal to the [log-likelihood function for multinomial observations](/P/mult-mle) at the value $p_0 = [1/k, \ldots, 1/k]$:
 
 $$ \label{eq:Mult-LME-m0}
 \begin{split}

P/mult-mle.md

@@ -40,7 +40,7 @@ $$ \label{eq:Mult-marg}
 X \sim \mathrm{Mult}(n,p) \quad \Rightarrow \quad X_j \sim \mathrm{Bin}(n, p_j) \quad \text{for all} \quad j = 1, \ldots, k \; .
 $$
 
-Thus, combining \eqref{eq:Mult} with \eqref{eq:Mult}, we have
+Thus, combining \eqref{eq:Mult} with \eqref{eq:Mult-marg}, we have
 
 $$ \label{eq:Mult-Bin}
 y_j \sim \mathrm{Bin}(n,p_j)

P/mult-pp.md

@@ -39,7 +39,7 @@ $$
 Then, the [posterior probability](/D/pmp) of the [alternative model](/D/h1) is given by
 
 $$ \label{eq:Mult-PP1}
-p(m_1|y) = 
+p(m_1|y) = \frac{1}{1 + k^{-n} \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{0j})}{\prod_{j=1}^k \Gamma(\alpha_{nj})}}
 $$
 
 where $\Gamma(x)$ is the gamma function and $\alpha_n$ are the [posterior hyperparameters for multinomial observations](/P/mult-post) which are functions of the [numbers of observations](/D/mult) $y_1, \ldots, y_k$.

P/ugkv-lbf.md

@@ -54,7 +54,7 @@ $$
 The LME of the alternative $m_1$ is equal to the [log model evidence for the univariate Gaussian with known variance](/P/ugkv-lme):
 
 $$ \label{eq:UGkv-LME-m1}
-\mathrm{LME}(m_1) = \frac{n}{2} \log\left( \frac{\tau}{2 \pi} \right) + \frac{1}{2} \log\left( \frac{\lambda_0}{\lambda_n} \right) - \frac{1}{2} \left( \tau y^\mathrm{T} y + \lambda_0 \mu_0^2 - \lambda_n \mu_n^2 \right) \; .
+\mathrm{LME}(m_1) = \log p(y|m_1) = \frac{n}{2} \log\left( \frac{\tau}{2 \pi} \right) + \frac{1}{2} \log\left( \frac{\lambda_0}{\lambda_n} \right) - \frac{1}{2} \left( \tau y^\mathrm{T} y + \lambda_0 \mu_0^2 - \lambda_n \mu_n^2 \right) \; .
 $$
 
 Because the null model $m_0$ has no free parameter, its [log model evidence](/D/lme) (logarithmized [marginal likelihood](/D/ml)) is equal to the [log-likelihood function for the univariate Gaussian with known variance](/P/ugkv-mle) at the value $\mu = 0$: