corrected some pages

Several small corrections were done to several proofs and definitions.

Author: JoramSoch

P/beta-mode.md

@@ -217,12 +217,12 @@ $$ \label{eq:beta-mode-p2b-s6}
 \end{split}
 $$
 
-Thus, $f''_X(x)$ is negative for $x = \frac{\alpha-1}{\alpha+\beta-2}$, demonstrating that this is a maximum. To summarize:
+Thus, $f'\'\_X(x)$ is negative for $x = \frac{\alpha-1}{\alpha+\beta-2}$, demonstrating that this is a maximum. To summarize:
 
 * If $\alpha < 1$ and $\beta < 1$, then $f_X(x)$ diverges at both ends and both values from the set $\left\lbrace 0, 1 \right\rbrace$ can be seen as the mode of $X$.
 
 * If $\alpha < 1$ or $\beta < 1$ (but not $\alpha < 1$ and $\beta < 1$), then the mode of $X$ is 0 or 1, because $f_X(x)$ tends towards infinity at $x = 0$ or $x = 1$.
 
 * If $\alpha = 1$ and $\beta = 1$, then $f_X(x)$ is constant and any value in the interval $\left[ 0,1 \right]$ can be seen as the mode of $X$.
 
-* If $\alpha \geq 1$ and $\beta \geq 1$ (but not $\alpha = 1$ and $\beta = 1$), then $0 < x =  < 1$ and $f'_X(x) = 0$ and $f''_X(x) < 0$, such that $f_X(x)$ reaches its machimum at $\mathrm{mode}(X) = \frac{\alpha-1}{\alpha+\beta-2}$.
+* If $\alpha \geq 1$ and $\beta \geq 1$ (but not $\alpha = 1$ and $\beta = 1$), then $0 < x =  < 1$ and $f'_X(x) = 0$ and $f'\'\_X(x) < 0$, such that $f_X(x)$ reaches its machimum at $\mathrm{mode}(X) = \frac{\alpha-1}{\alpha+\beta-2}$.

P/betabin-mome.md

@@ -108,15 +108,15 @@ m_2 &= \frac{\alpha \left( n + n m_1 - m_1 \right) + n m_1}{\alpha \frac{n}{m_1}
 m_2 \left( \frac{\alpha n}{m_1} + 1 \right) &= \alpha \left( n + n m_1 - m_1 \right) + n m_1 \\
 \alpha \left( n \frac{m_2}{m_1} - (n + n m_1 - m_1) \right) &= n m_1 - m_2 \\
 \alpha \left( n \left( \frac{m_2}{m_1} - m_1 - 1 \right) + m_1 \right) &= n m_1 - m_2 \\
-\alpha &= \frac{n m_1 - m_2}{n \left( \frac{m_2}{m_1} - m_1 - 1 \right) + m_1} \; .
+\hat{\alpha} &= \frac{n m_1 - m_2}{n \left( \frac{m_2}{m_1} - m_1 - 1 \right) + m_1} \; .
 \end{split}
 $$
 
 Plugging this into equation \eqref{eq:beta-as-alpha}, one obtains for $\beta$:
 
 $$ \label{eq:binbeta-mome-beta}
 \begin{split}
-\beta &= \alpha \left( \frac{n}{m_1} - 1 \right) \\
+\beta &= \hat{\alpha} \left( \frac{n}{m_1} - 1 \right) \\
 \beta &= \left( \frac{n m_1 - m_2}{n \left( \frac{m_2}{m_1} - m_1 - 1 \right) + m_1} \right) \left( \frac{n}{m_1} - 1 \right) \\
 \beta &= \frac{n^2 - n m_1 - n \frac{m_2}{m_1} + m_2}{n \left( \frac{m_2}{m_1} - m_1 - 1 \right) + m_1} \\
 \hat{\beta} &= \frac{\left( n - m_1 \right)\left( n - \frac{m_2}{m_1} \right)}{n \left( \frac{m_2}{m_1} - m_1 - 1 \right) + m_1} \; .

P/blr-map.md

@@ -45,7 +45,7 @@ $$
 where $n$ is the [number of data points](/D/mlr).
 
 
-**Proof:** Given the [prior distribution](/D/prior) in \eqref{eq:GLM-NG-prior}, the [posterior distribution](/D/post) for [multiple linear regression](/D/mlr) [is also a normal-gamma distribution](/P/blr-post)
+**Proof:** Given the [generative model](/D/gm) in \eqref{eq:GLM} and the [prior distribution](/D/prior) in \eqref{eq:GLM-NG-prior}, the [posterior distribution](/D/post) for [multiple linear regression](/D/mlr) [is also a normal-gamma distribution](/P/blr-post)
 
 $$ \label{eq:GLM-NG-post}
 p(\beta,\tau|y) = \mathcal{N}(\beta; \mu_n, (\tau \Lambda_n)^{-1}) \cdot \mathrm{Gam}(\tau; a_n, b_n)

P/chi2-mom.md P/chi2-momraw.mdP/chi2-mom.md renamed to P/chi2-momraw.md

@@ -7,11 +7,11 @@ affiliation: "KU Leuven"
 e_mail: "kenneth.petrykowski@gmail.com"
 date: 2020-10-13 01:30:00
 
-title: "Moments of the chi-squared distribution"
+title: "Raw moments of the chi-squared distribution"
 chapter: "Probability Distributions"
 section: "Univariate continuous distributions"
 topic: "Chi-squared distribution"
-theorem: "Moments"
+theorem: "Raw moments"
 
 sources:
   - authors: "Robert V. Hogg, Joseph W. McKean, Allen T. Craig"
@@ -22,7 +22,7 @@ sources:
     url: "https://www.pearson.com/store/p/introduction-to-mathematical-statistics/P100000843744"
 
 proof_id: "P175"
-shortcut: "chi2-mom"
+shortcut: "chi2-momraw"
 username: "kjpetrykowski"
 ---
 
@@ -33,16 +33,16 @@ $$ \label{eq:chi2}
 X \sim \chi^{2}(k) \; .
 $$
 
-Then, if $m > -k/2$, the moment $\mathrm{E}(X^{m})$ exists and is equal to:
+Then, if $m > -k/2$, the [raw moment](/D/mom-raw) $\mathrm{E}(X^{m})$ exists and is equal to:
 
-$$ \label{eq:chi2-mom}
+$$ \label{eq:chi2-momraw}
 \mathrm{E}(X^{m}) = 2^m \frac{\Gamma\left( \frac{k}{2}+m \right)}{\Gamma\left( \frac{k}{2} \right)} \; .
 $$
 
 
 **Proof:** Combining the [definition of the raw moment](/D/mom-raw) with the [probability density function of the chi-squared distribution](/P/chi2-pdf), we have:
 
-$$ \label{eq:chi2-mom-int}
+$$ \label{eq:chi2-momraw-int}
 \begin{split}
 \mathrm{E}(X^{m}) &= \int_{0}^{\infty} x^m \frac{1}{2^{k/2} \Gamma\left( \frac{k}{2} \right)} \, x^{k/2-1} \, e^{-x/2} \, \mathrm{d}x \\
 &= \frac{1}{2^{k/2} \Gamma\left( \frac{k}{2} \right)} \int_{0}^{\infty} x^{(k/2)+m-1} \, e^{-x/2} \, \mathrm{d}x \; .
@@ -51,7 +51,7 @@ $$
 
 Now, we substitute $u = x/2$, such that $x = 2u$. As a result, we obtain:
 
-$$ \label{eq:chi2-mom-int-u}
+$$ \label{eq:chi2-momraw-int-u}
 \begin{split}
 \mathrm{E}(X^{m}) &= \frac{1}{2^{k/2} \Gamma\left( \frac{k}{2} \right)} \int_{0}^{\infty} 2^{(k/2)+m-1} \, u^{(k/2)+m-1} \, e^{-u} \, \mathrm{d}(2u) \\
 &= \frac{2^{(k/2)+m}}{2^{k/2} \Gamma\left( \frac{k}{2} \right)} \int_{0}^{\infty} u^{(k/2)+m-1} \, e^{-u} \, \mathrm{d}u \\
@@ -65,4 +65,4 @@ $$ \label{eq:gam-fct}
 \Gamma(x) = \int_{0}^{\infty} t^{x-1} \, e^{-t} \, \mathrm{d}t, \; z > 0 \; ,
 $$
 
-this leads to the desired result when $m > -k/2$. Observe that, if $m$ is a nonnegative integer, then $m > -k/2$ is always true. Therefore, all [moments](/D/mom) of a [chi-squared distribution](/D/chi2) exist and the $m$-th raw moment is given by the equation above.
+this leads to the desired result when $m > -k/2$. Observe that, if $m$ is a non-negative integer, then $m > -k/2$ is always true. Therefore, all [moments](/D/mom) of a [chi-squared distribution](/D/chi2) exist and the $m$-th raw moment is given by the equation above.

P/dir-kl.md

@@ -27,12 +27,12 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $x$ be an $1 \times k$ [random vector](/D/rvec). Assume two [Dirichlet distributions](/D/dir) $P$ and $Q$ specifying the probability distribution of $x$ as
+**Theorem:** Let $X$ be an $1 \times k$ [random vector](/D/rvec). Assume two [Dirichlet distributions](/D/dir) $P$ and $Q$ specifying the probability distribution of $X$ as
 
 $$ \label{eq:dirs}
 \begin{split}
-P: \; x &\sim \mathrm{Dir}(\alpha_1) \\
-Q: \; x &\sim \mathrm{Dir}(\alpha_2) \; .
+P: \; X &\sim \mathrm{Dir}(\alpha_1) \\
+Q: \; X &\sim \mathrm{Dir}(\alpha_2) \; .
 \end{split}
 $$
 
@@ -74,7 +74,7 @@ $$
 Using the [expected value of a logarithmized Dirichlet variate](/P/dir-logmean)
 
 $$ \label{eq:dir-logmean}
-x \sim \mathrm{Dir}(\alpha) \quad \Rightarrow \quad \left\langle \ln x_i \right\rangle = \psi(\alpha_i) - \psi\left(\sum_{i=1}^{k} \alpha_i\right) \; ,
+X \sim \mathrm{Dir}(\alpha) \quad \Rightarrow \quad \left\langle \ln x_i \right\rangle = \psi(\alpha_i) - \psi\left(\sum_{i=1}^{k} \alpha_i\right) \; ,
 $$
 
 the Kullback-Leibler divergence from \eqref{eq:dir-KL-s2} becomes:

P/dir-mle.md

@@ -108,7 +108,7 @@ $$ \label{eq:Dir-dLLdaj-0}
 \end{split}
 $$
 
-In the following, we will use a fixed-point iteration to maximize $\mathrm{LL}(\alpha)$. Given an initial guess for $\alpha$, we construct a lower bound on the likelihood function \eqref{eq:Dir-LL-der} which is tight at $\alpha$. The maximum of this bound is computed and it becomes the new guess. Because the [Dirichlet distribution](/D/dir) belongs to the [exponential family](/D/dist-expfam), the log-likelihood function is convex in $\alpha$ ánd the maximum is the only stationary point, such that the procedure is guaranteed to converge to the maximum.
+In the following, we will use a fixed-point iteration to maximize $\mathrm{LL}(\alpha)$. Given an initial guess for $\alpha$, we construct a lower bound on the likelihood function \eqref{eq:Dir-LL-der} which is tight at $\alpha$. The maximum of this bound is computed and it becomes the new guess. Because the [Dirichlet distribution](/D/dir) belongs to the [exponential family](/D/dist-expfam), the log-likelihood function is convex in $\alpha$ and the maximum is the only stationary point, such that the procedure is guaranteed to converge to the maximum.
 
 In our case, we use a bound on the gamma function
 

P/lognorm-med.md

@@ -52,7 +52,7 @@ $$ \label{eq:erf}
 \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp(-t^2) \, \mathrm{d}t \; .
 $$
 
-Thus, the inverse CDF is
+Thus, the inverse cumulative distribution function is
 
 $$ \label{eq:lognorm-cdf-inv}
 \begin{split}

P/lognorm-var.md

@@ -84,7 +84,7 @@ $$ \label{eq:second-moment-2}
 \end{split}
 $$
 
-Now multiplying by $\exp \left( 2 \sigma^2 \right)$ and $\exp \left(- 2 \sigma^2 \right)$, this becomes:
+Now multiplying the integrand by $\exp \left( 2 \sigma^2 \right)$ and $\exp \left(- 2 \sigma^2 \right)$, this becomes:
 
 $$ \label{eq:second-moment-3}
 \begin{split}

P/mom-mgf.md

@@ -39,7 +39,7 @@ where $n$ is a positive integer and $M_X^{(n)}(t)$ is the $n$-th derivative of $
 **Proof:** Using the [definition of the moment-generating function](/D/mgf), we can write:
 
 $$ \label{eq:mom-mgf-s1}
-M_X^{(n)}(t) = \frac{\mathrm{d}^n}{\mathrm{d}t^n} \mathrm{E}(e^{tX}) \; .
+M_X^{(n)}(t) = \frac{\mathrm{d}^n}{\mathrm{d}t^n} \mathrm{E}\left( e^{tX} \right) \; .
 $$
 
 Using the power series expansion of the exponential function

P/poiss-lme.md

@@ -39,7 +39,7 @@ $$ \label{eq:Poiss-LME}
 \log p(y|m) = - \sum_{i=1}^n \log y_i ! + \log \Gamma(a_n) - \log \Gamma(a_0) + a_0 \log b_0 - a_n \log b_n \; .
 $$
 
-and the [posterior hyperparameters](/D/post) are given by
+where the [posterior hyperparameters](/D/post) are given by
 
 $$ \label{eq:Poiss-post-par}
 \begin{split}