corrected some pages

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

Author: JoramSoch

D/ind-cond.md

@@ -39,7 +39,7 @@ $$
 where $p(x_1, \ldots, x_n \vert y)$ are the [joint (conditional) probabilities](/D/prob-joint) of $X_1, \ldots, X_n$ given $Y$ and $p(x_i)$ are the [marginal (conditional) probabilities](/D/prob-marg) of $X_i$ given $Y$.
 
 <br>
-2) A set of [random variables](/D/rvar) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called conditionally independent given the random variable $Y$ with possible values $\mathcal{Y}$, if
+2) A set of [continuous random variables](/D/rvar-cont) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called conditionally independent given the random variable $Y$ with possible values $\mathcal{Y}$, if
 
 $$ \label{eq:cond-ind-F}
 F_{X_1,\ldots,X_n|Y=y}(x_1,\ldots,x_n) = \prod_{i=1}^{n} F_{X_i|Y=y}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i \quad \text{and all} \; y \in \mathcal{Y}

D/qf.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a [random variable](/D/rvar) with the [cumulative distribution function](/D/cdf) (CDF) $F_X(x)$. Then, the function $Q_X(p): [0,1] \to \mathbb{R}$ which is the inverse CDF is the quantile function (QF) of $X$. More precisly, the QF is the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
+**Definition:** Let $X$ be a [random variable](/D/rvar) with the [cumulative distribution function](/D/cdf) (CDF) $F_X(x)$. Then, the function $Q_X(p): [0,1] \to \mathbb{R}$ which is the inverse CDF is the quantile function (QF) of $X$. More precisely, the QF is the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
 
 $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .

D/rvar.md

@@ -33,4 +33,4 @@ username: "JoramSoch"
 
 * formally, as a [measurable function](/D/meas-fct) $X$ defined on a [probability space](/D/prob-spc) $(\Omega, \mathcal{F}, P)$ that maps from a sample space $\Omega$ to the real numbers $\mathbb{R}$ using an event space $\mathcal{F}$ and a [probability function](/D/pmf) $P$;
 
-* more broadly, as any random quantity $X$ such as a [random scalar](/D/rvar), a [random vector](/D/rvec) or a [random matrix](/D/rmat).
+* more broadly, as any random quantity $X$ such as a [random event](/D/reve), a [random scalar](/D/rvar), a [random vector](/D/rvec) or a [random matrix](/D/rmat).

I/Table_of_Contents.md

@@ -56,11 +56,11 @@ title: "Table of Contents"
    &emsp;&ensp; 1.4.12. **[Cumulative distribution function of continuous random variable](/P/cdf-pdf)** <br>
    &emsp;&ensp; 1.4.13. *[Quantile function](/D/qf)* <br>
    &emsp;&ensp; 1.4.14. **[Quantile function in terms of cumulative distribution function](/P/qf-cdf)** <br>
-   &emsp;&ensp; 1.4.11. *[Moment-generating function](/D/mgf)* <br>
-   &emsp;&ensp; 1.4.12. **[Moment-generating function of linear transformation](/P/mgf-ltt)** <br>
-   &emsp;&ensp; 1.4.13. **[Moment-generating function of linear combination](/P/mgf-lincomb)** <br>
-   &emsp;&ensp; 1.4.14. *[Cumulant-generating function](/D/cgf)* <br>
-   &emsp;&ensp; 1.4.15. *[Probability-generating function](/D/pgf)* <br>
+   &emsp;&ensp; 1.4.15. *[Moment-generating function](/D/mgf)* <br>
+   &emsp;&ensp; 1.4.16. **[Moment-generating function of linear transformation](/P/mgf-ltt)** <br>
+   &emsp;&ensp; 1.4.17. **[Moment-generating function of linear combination](/P/mgf-lincomb)** <br>
+   &emsp;&ensp; 1.4.18. *[Cumulant-generating function](/D/cgf)* <br>
+   &emsp;&ensp; 1.4.19. *[Probability-generating function](/D/pgf)* <br>
 
    1.5. Expected value <br>
    &emsp;&ensp; 1.5.1. *[Definition](/D/mean)* <br>
@@ -479,17 +479,17 @@ title: "Table of Contents"
    &emsp;&ensp; 3.2.2. **[Derivation](/P/lfe-der)** <br>
    &emsp;&ensp; 3.2.3. **[Calculation from log model evidences](/P/lfe-lme)** <br>
 
-   3.3. Bayes factor <br>
-   &emsp;&ensp; 3.3.1. *[Definition](/D/bf)* <br>
-   &emsp;&ensp; 3.3.2. **[Transitivity](/P/bf-trans)** <br>
-   &emsp;&ensp; 3.3.3. **[Computation using Savage-Dickey Density Ratio](/P/bf-sddr)** <br>
-   &emsp;&ensp; 3.3.4. **[Computation using Encompassing Prior Method](/P/bf-ep)** <br>
-   &emsp;&ensp; 3.3.5. *[Encompassing model](/D/encm)* <br>
-   
-   3.4. Log Bayes factor <br>
-   &emsp;&ensp; 3.4.1. *[Definition](/D/lbf)* <br>
-   &emsp;&ensp; 3.4.2. **[Derivation](/P/lbf-der)** <br>
-   &emsp;&ensp; 3.4.3. **[Calculation from log model evidences](/P/lbf-lme)** <br>
+   3.3. Log Bayes factor <br>
+   &emsp;&ensp; 3.3.1. *[Definition](/D/lbf)* <br>
+   &emsp;&ensp; 3.3.2. **[Derivation](/P/lbf-der)** <br>
+   &emsp;&ensp; 3.3.3. **[Calculation from log model evidences](/P/lbf-lme)** <br>
+   
+   3.4. Bayes factor <br>
+   &emsp;&ensp; 3.4.1. *[Definition](/D/bf)* <br>
+   &emsp;&ensp; 3.4.2. **[Transitivity](/P/bf-trans)** <br>
+   &emsp;&ensp; 3.4.3. **[Computation using Savage-Dickey Density Ratio](/P/bf-sddr)** <br>
+   &emsp;&ensp; 3.4.4. **[Computation using Encompassing Prior Method](/P/bf-ep)** <br>
+   &emsp;&ensp; 3.4.5. *[Encompassing model](/D/encm)* <br>
 
    3.5. Posterior model probability <br>
    &emsp;&ensp; 3.5.1. *[Definition](/D/pmp)* <br>

P/beta-cdf.md

@@ -54,7 +54,7 @@ $$ \label{eq:beta-cdf-app}
 \begin{split}
 F_X(x) &= \int_{0}^{x} \mathrm{Bet}(z; \alpha, \beta) \, \mathrm{d}z \\
 &= \int_{0}^{x} \frac{1}{\mathrm{B}(\alpha, \beta)} \, z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \\
-&= \frac{1}{B(x;a,b)} \int_{0}^{x} z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \; .
+&= \frac{1}{\mathrm{B}(\alpha, \beta)} \int_{0}^{x} z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \; .
 \end{split}
 $$
 

P/cuni-qf.md

@@ -57,7 +57,7 @@ $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
 $$
 
-Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, it holds that
+Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, [it holds that](/P/qf-cdf)
 
 $$ \label{eq:exp-qf-s1}
 Q_X(p) = F_X^{-1}(x) \; .

P/exp-qf.md

@@ -56,7 +56,7 @@ $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
 $$
 
-Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, it holds that
+Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, [it holds that](/P/qf-cdf)
 
 $$ \label{eq:exp-qf-s1}
 Q_X(p) = F_X^{-1}(x) \; .

P/gam-cdf.md

@@ -70,7 +70,7 @@ $$
 
 With the definition of the lower incomplete gamma function
 
-$$ \label{eq:li-gam-fct}
+$$ \label{eq:low-inc-gam-fct}
 \gamma(s,x) = \int_{0}^{x} t^{s-1} \exp[-t] \, \mathrm{d}t \; ,
 $$
 

P/gam-kl.md

@@ -26,12 +26,12 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $x$ be a [random variable](/D/rvar). Assume two [gamma distributions](/D/gam) $P$ and $Q$ specifying the probability distribution of $x$ as
+**Theorem:** Let $X$ be a [random variable](/D/rvar). Assume two [gamma distributions](/D/gam) $P$ and $Q$ specifying the probability distribution of $X$ as
 
 $$ \label{eq:gams}
 \begin{split}
-P: \; x &\sim \mathrm{Gam}(a_1, b_1) \\
-Q: \; x &\sim \mathrm{Gam}(a_2, b_2) \; . \\
+P: \; X &\sim \mathrm{Gam}(a_1, b_1) \\
+Q: \; X &\sim \mathrm{Gam}(a_2, b_2) \; . \\
 \end{split}
 $$
 

P/lme-anc.md

@@ -41,10 +41,10 @@ $$ \label{eq:LME}
 \mathrm{LME}(m) = \mathrm{Acc}(m) - \mathrm{Com}(m)
 $$
 
-where the accuracy term is the posterior expectation of the log-[likelihood function](/D/lf)
+where the accuracy term is the [posterior](/D/post) [expectation](/D/mean-lotus) of the [log-likelihood function](/D/llf)
 
 $$ \label{eq:Acc}
-\mathrm{Acc}(m) = \left\langle p(y|\theta,m) \right\rangle_{p(\theta|y,m)}
+\mathrm{Acc}(m) = \left\langle \log p(y|\theta,m) \right\rangle_{p(\theta|y,m)}
 $$
 
 and the complexity penalty is the [Kullback-Leibler divergence](/D/kl) of [posterior](/D/post) from [prior](/D/prior)
@@ -54,7 +54,7 @@ $$ \label{eq:Com}
 $$
 
 
-**Proof:** We consider Bayesian inference on data $y$ using model $m$ with parameters $\theta$. Then, [Bayes' theorem](/P/bayes-th) makes a statement about the posterior distribution, i.e. the probability of parameters, given the data and the model:
+**Proof:** We consider Bayesian inference on [data](/D/data) $y$ using [model](/D/gm) $m$ with parameters $\theta$. Then, [Bayes' theorem](/P/bayes-th) makes a statement about the [posterior distribution](/D/post), i.e. the probability of parameters, given the data and the model:
 
 $$ \label{eq:AnC-s1}
 p(\theta|y,m) = \frac{p(y|\theta,m) \, p(\theta|m)}{p(y|m)} \; .
@@ -81,7 +81,7 @@ $$
 By definition, the left-hand side is the log model evidence and the terms on the right-hand side correspond to the posterior expectation of the log-likelihood function and the Kullback-Leibler divergence of posterior from prior
 
 $$ \label{eq:LME-AnC}
-\mathrm{LME}(m) = \left\langle p(y|\theta,m) \right\rangle_{p(\theta|y,m)} - \mathrm{KL} \left[ p(\theta|y,m) \, || \, p(\theta|m) \right]
+\mathrm{LME}(m) = \left\langle \log p(y|\theta,m) \right\rangle_{p(\theta|y,m)} - \mathrm{KL} \left[ p(\theta|y,m) \, || \, p(\theta|m) \right]
 $$
 
 which proofs the partition given by \eqref{eq:LME}.