Merge pull request #88 from StatProofBook/master

update to master

Author: JoramSoch

I/PbA.md

@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (311 proofs)
+### JoramSoch (313 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -175,6 +175,8 @@ title: "Proof by Author"
 - [Moment-generating function of the normal distribution](/P/norm-mgf)
 - [Monotonicity of probability](/P/prob-mon)
 - [Monotonicity of the expected value](/P/mean-mono)
+- [Multiple linear regression is a special case of the general linear model](/P/mlr-glm)
+- [Multivariate normal distribution is a special case of matrix-normal distribution](/P/mvn-matn)
 - [Necessary and sufficient condition for independence of multivariate normal random variables](/P/mvn-ind)
 - [Non-invariance of the differential entropy under change of variables](/P/dent-noninv)
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)

I/PbN.md

@@ -334,3 +334,5 @@ title: "Proof by Number"
 | P326 | lognorm-qf | [Quantile function of the log-normal distribution](/P/lognorm-qf) | majapavlo | 2022-07-09 |
 | P327 | nw-mean | [Mean of the normal-Wishart distribution](/P/nw-mean) | JoramSoch | 2022-07-14 |
 | P328 | gam-wish | [Gamma distribution is a special case of Wishart distribution](/P/gam-wish) | JoramSoch | 2022-07-14 |
+| P329 | mlr-glm | [Multiple linear regression is a special case of the general linear model](/P/mlr-glm) | JoramSoch | 2022-07-21 |
+| P330 | mvn-matn | [Multivariate normal distribution is a special case of matrix-normal distribution](/P/mvn-matn) | JoramSoch | 2022-07-31 |

I/PbT.md

@@ -214,6 +214,8 @@ title: "Proof by Topic"
 - [Moments of the chi-squared distribution](/P/chi2-mom)
 - [Monotonicity of probability](/P/prob-mon)
 - [Monotonicity of the expected value](/P/mean-mono)
+- [Multiple linear regression is a special case of the general linear model](/P/mlr-glm)
+- [Multivariate normal distribution is a special case of matrix-normal distribution](/P/mvn-matn)
 
 ### N
 

I/PwS.md

@@ -65,6 +65,7 @@ title: "Proofs without Source"
 - [Mode of the continuous uniform distribution](/P/cuni-med)
 - [Mode of the exponential distribution](/P/exp-mode)
 - [Mode of the normal distribution](/P/norm-mode)
+- [Multivariate normal distribution is a special case of matrix-normal distribution](/P/mvn-matn)
 - [Necessary and sufficient condition for independence of multivariate normal random variables](/P/mvn-ind)
 - [Normal-gamma distribution is a special case of normal-Wishart distribution](/P/ng-nw)
 - [Ordinary least squares for simple linear regression](/P/slr-ols2)

P/glm-mle.md

@@ -60,7 +60,7 @@ Substituting $V^{-1}$ by the precision matrix $P$ to ease notation, we have:
 
 $$ \label{eq:GLM-LL2}
 \begin{split}
-\mathrm{LL}(B,\Sigma) = &- \frac{nv}{2} \log(2\pi) - \frac{n}{2} \log(|\Sigma|) - \frac{v}{2} \log(|V|) \\
+\mathrm{LL}(B,\Sigma) = &- \frac{nv}{2} \log(2\pi) - \frac{n}{2} \log(|\Sigma|) + \frac{v}{2} \log(|P|) \\
 &- \frac{1}{2} \, \mathrm{tr}\left[ \Sigma^{-1} \left( Y^\mathrm{T} P Y - Y^\mathrm{T} P X B - B^\mathrm{T} X^\mathrm{T} P Y + B^\mathrm{T} X^\mathrm{T} P X B \right) \right] \; .\\
 \end{split}
 $$

P/mblr-lme.md

@@ -37,7 +37,7 @@ Then, the [log model evidence](/D/lme) for this model is
 
 $$ \label{eq:GLM-NW-LME}
 \begin{split}
-\log p(y|m) = & \frac{v}{2} \log |P| - \frac{nv}{2} \log (2 \pi)  + \frac{v}{2} \log |\Lambda_0| - \frac{v}{2} \log |\Lambda_n| + \\
+\log p(Y|m) = & \frac{v}{2} \log |P| - \frac{nv}{2} \log (2 \pi)  + \frac{v}{2} \log |\Lambda_0| - \frac{v}{2} \log |\Lambda_n| + \\
 & \frac{\nu_0}{2} \log\left| \frac{1}{2} \Omega_0 \right| - \frac{\nu_n}{2} \log\left| \frac{1}{2} \Omega_n \right| + \log \Gamma_v \left( \frac{\nu_n}{2} \right) - \log \Gamma_v \left( \frac{\nu_0}{2} \right)
 \end{split}
 $$
@@ -124,7 +124,7 @@ Thus, the [log model evidence](/D/lme) of this model is given by
 
 $$ \label{eq:GLM-NW-LME-s6}
 \begin{split}
-\log p(y|m) = & \frac{v}{2} \log |P| - \frac{nv}{2} \log (2 \pi)  + \frac{v}{2} \log |\Lambda_0| - \frac{v}{2} \log |\Lambda_n| + \\
+\log p(Y|m) = & \frac{v}{2} \log |P| - \frac{nv}{2} \log (2 \pi)  + \frac{v}{2} \log |\Lambda_0| - \frac{v}{2} \log |\Lambda_n| + \\
 & \frac{\nu_0}{2} \log\left| \frac{1}{2} \Omega_0 \right| - \frac{\nu_n}{2} \log\left| \frac{1}{2} \Omega_n \right| + \log \Gamma_v \left( \frac{\nu_n}{2} \right) - \log \Gamma_v \left( \frac{\nu_0}{2} \right) \; .
 \end{split}
 $$

P/mvn-matn.md

@@ -14,6 +14,12 @@ topic: "Multivariate normal distribution"
 theorem: "Special case of matrix-normal distribution"
 
 sources:
+  - authors: "Wikipedia"
+    year: 2022
+    title: "Matrix normal distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2022-07-31"
+    url: "https://en.wikipedia.org/wiki/Matrix_normal_distribution"
 
 proof_id: "P330"
 shortcut: "mvn-matn"
@@ -32,11 +38,11 @@ $$
 
 Setting $p = 1$, $X = x$, $M = \mu$, $U = \Sigma$ and $V = 1$, we obtain
 
-\begin{equation} \label{eq:exp-pdf}
+$$ \label{eq:exp-pdf}
 \begin{split}
 \mathcal{MN}(x; \mu, \Sigma, 1) &= \frac{1}{\sqrt{(2\pi)^{n} |1|^n |\Sigma|^1}} \cdot \exp\left[-\frac{1}{2} \mathrm{tr}\left( 1^{-1} (x-\mu)^\mathrm{T} \, \Sigma^{-1} (x-\mu) \right) \right] \\
 &= \frac{1}{\sqrt{(2\pi)^{n} |\Sigma|}} \cdot \exp\left[-\frac{1}{2} (x-\mu)^\mathrm{T} \, \Sigma^{-1} (x-\mu) \right]
 \end{split}
-\end{equation}
+$$
 
 which is equivalent to the [probability density function of the multivariate normal distribution](/P/mvn-pdf).