corrected some pages

Several small corrections were done to several proofs and definitions.

Author: JoramSoch

D/map.md

@@ -33,7 +33,7 @@ $$ \label{eq:post}
 \theta|y \sim \mathcal{D}(\phi) \; .
 $$
 
-Then, the value of $\theta$ at which the [posterior density](/D/post) attains its maximum is called the "maximum-a-posteriori estimate" or "MAP estimate" of $\theta$:
+Then, the value of $\theta$ at which the [posterior density](/D/post) attains its maximum is called the "maximum-a-posteriori estimate", "MAP estimate" or "posterior mode" of $\theta$:
 
 $$ \label{eq:prior-pdf}
 \hat{\theta}_\mathrm{MAP} = \operatorname*{arg\,max}_\theta \mathcal{D}(\theta; \phi) \; .

D/mle.md

@@ -21,7 +21,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let there be a [generative model](/D/gm) $m$ describing measured data $y$ using model parameters $\theta$. Then, the parameter values maximizing the [likelihood function](/D/lf) or [log-likelihood function](/D/llf) are called maximum likelihood estimates of $\theta$:
+**Definition:** Let there be a [generative model](/D/gm) $m$ describing measured data $y$ using model parameters $\theta$. Then, the parameter values maximizing the [likelihood function](/D/lf) or [log-likelihood function](/D/llf) are called "maximum likelihood estimates" of $\theta$:
 
 $$ \label{eq:mle}
 \hat{\theta} = \operatorname*{arg\,max}_\theta \mathcal{L}_m(\theta) = \operatorname*{arg\,max}_\theta \mathrm{LL}_m(\theta) \; .

P/bin-map.md

@@ -35,7 +35,7 @@ $$
 
 Then, the [maximum-a-posteriori estimate](/D/map) of $p$ is
 
-$$ \label{eq:Bin-MLE}
+$$ \label{eq:Bin-MAP}
 \hat{p}_\mathrm{MAP} = \frac{\alpha_0+y-1}{\alpha_0+\beta_0+n-2} \; .
 $$
 
@@ -57,13 +57,13 @@ $$
 
 The [mode of the beta distribution](/P/beta-mode) is given by:
 
-$$ \label{eq:beta-mode}
+$$ \label{eq:Beta-mode}
 X \sim \mathrm{Bet}(\alpha, \beta) \quad \Rightarrow \quad \mathrm{mode}(X) = \frac{\alpha-1}{\alpha+\beta-2} \; .
 $$
 
-Applying \eqref{eq:beta-mode} to \eqref{eq:Bin-post} with \eqref{eq:Bin-post-par}, the [maximum-a-posteriori estimate](/D/map) of $p$ follows as:
+Applying \eqref{eq:Beta-mode} to \eqref{eq:Bin-post} with \eqref{eq:Bin-post-par}, the [maximum-a-posteriori estimate](/D/map) of $p$ follows as:
 
-$$ \label{eq:Bin-MAP}
+$$ \label{eq:Bin-MAP-qed}
 \begin{split}
 \hat{p}_\mathrm{MAP} &= \frac{\alpha_n-1}{\alpha_n+\beta_n-2} \\
 &\overset{\eqref{eq:Bin-post-par}}{=} \frac{\alpha_0+y-1}{\alpha_0+y+\beta_0+(n-y)-2} \\

P/mult-map.md

@@ -71,7 +71,8 @@ $$
 Since $y_1 + \ldots + y_k = n$ [by definition](/D/mult-data), this becomes
 
 $$ \label{eq:Mult-MAP-s2}
-\hat{p}_{i,\mathrm{MAP}} = \frac{\alpha_{0i} + y_i - 1}{\sum_j \alpha_{0j} + n - k} \end{equation}
+\hat{p}_{i,\mathrm{MAP}} = \frac{\alpha_{0i} + y_i - 1}{\sum_j \alpha_{0j} + n - k}
+$$
 
 which, using the $1 \times k$ [vectors](/D/mult-data) $y$, $p$ and $\alpha_0$, can be written as: