corrected some pages

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

Author: JoramSoch

P/blr-lme.md

@@ -33,7 +33,7 @@ $$ \label{eq:GLM}
 m: \; y = X \beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V)
 $$
 
-be a [linear regression model](/D/mlr) with measured $n \times 1$ data vector $y$, known $n \times p$ design matrix $X$, known $n \times n$ covariance structure $V$ and unknown $p \times 1$ regression coefficients $\beta$ and noise variance $\sigma^2$.  Moreover, assume a [normal-gamma prior distribution](/P/blr-prior) over the model parameters $\beta$ and $\tau = 1/\sigma^2$:
+be a [linear regression model](/D/mlr) with measured $n \times 1$ data vector $y$, known $n \times p$ design matrix $X$, known $n \times n$ covariance structure $V$ as well as unknown $p \times 1$ regression coefficients $\beta$ and unknown noise variance $\sigma^2$. Moreover, assume a [normal-gamma prior distribution](/P/blr-prior) over the model parameters $\beta$ and $\tau = 1/\sigma^2$:
 
 $$ \label{eq:GLM-NG-prior}
 p(\beta,\tau) = \mathcal{N}(\beta; \mu_0, (\tau \Lambda_0)^{-1}) \cdot \mathrm{Gam}(\tau; a_0, b_0) \; .
@@ -117,13 +117,13 @@ $$
 Using the [probability density function of the gamma distribution](/P/gam-pdf), we can rewrite this as
 
 $$\label{eq:GLM-NG-LME-s4}
-\int p(y,\beta,\tau) \, \mathrm{d}\beta = \; \sqrt{\frac{|P|}{(2 \pi)^n}} \, \sqrt{\frac{|\Lambda_0|}{|\Lambda_n|}} \, \frac{ {b_0}^{a_0}}{\Gamma(a_0)} \, \frac{\Gamma(a_n)}{ {b_n}^{a_n}} \, \mathrm{Gam}(\tau; a_n, b_n) \; .
+\int p(y,\beta,\tau) \, \mathrm{d}\beta = \sqrt{\frac{|P|}{(2 \pi)^n}} \, \sqrt{\frac{|\Lambda_0|}{|\Lambda_n|}} \, \frac{ {b_0}^{a_0}}{\Gamma(a_0)} \, \frac{\Gamma(a_n)}{ {b_n}^{a_n}} \, \mathrm{Gam}(\tau; a_n, b_n) \; .
 $$
 
 Finally, $\tau$ can also be integrated out:
 
 $$ \label{eq:GLM-NG-LME-s5}
-\iint p(y,\beta,\tau) \, \mathrm{d}\beta \, \mathrm{d}\tau = \; \sqrt{\frac{|P|}{(2 \pi)^n}} \, \sqrt{\frac{|\Lambda_0|}{|\Lambda_n|}} \, \frac{\Gamma(a_n)}{\Gamma(a_0)} \, \frac{ {b_0}^{a_0}}{ {b_n}^{a_n}} = p(y|m) \; .
+\iint p(y,\beta,\tau) \, \mathrm{d}\beta \, \mathrm{d}\tau = \sqrt{\frac{|P|}{(2 \pi)^n}} \, \sqrt{\frac{|\Lambda_0|}{|\Lambda_n|}} \, \frac{\Gamma(a_n)}{\Gamma(a_0)} \, \frac{ {b_0}^{a_0}}{ {b_n}^{a_n}} = p(y|m) \; .
 $$
 
 Thus, the [log model evidence](/D/lme) of this model is given by

P/blr-post.md

@@ -33,7 +33,7 @@ $$ \label{eq:GLM}
 y = X \beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V)
 $$
 
-be a [linear regression model](/D/mlr) with measured $n \times 1$ data vector $y$, known $n \times p$ design matrix $X$, known $n \times n$ [covariance structure](/D/covmat) $V$ and unknown $p \times 1$ regression coefficients $\beta$ and noise variance $\sigma^2$.  Moreover, assume a [normal-gamma prior distribution](/P/blr-prior) over the model parameters $\beta$ and $\tau = 1/\sigma^2$:
+be a [linear regression model](/D/mlr) with measured $n \times 1$ data vector $y$, known $n \times p$ design matrix $X$, known $n \times n$ covariance structure $V$ as well as unknown $p \times 1$ regression coefficients $\beta$ and unknown noise variance $\sigma^2$. Moreover, assume a [normal-gamma prior distribution](/P/blr-prior) over the model parameters $\beta$ and $\tau = 1/\sigma^2$:
 
 $$ \label{eq:GLM-NG-prior}
 p(\beta,\tau) = \mathcal{N}(\beta; \mu_0, (\tau \Lambda_0)^{-1}) \cdot \mathrm{Gam}(\tau; a_0, b_0) \; .

P/blr-prior.md

@@ -33,7 +33,7 @@ $$ \label{eq:GLM}
 y = X \beta + \varepsilon, \; \varepsilon \sim \mathcal{N}(0, \sigma^2 V)
 $$
 
-be a [linear regression model](/D/mlr) with measured $n \times 1$ data vector $y$, known $n \times p$ design matrix $X$, known $n \times n$ covariance structure $V$ and unknown $p \times 1$ regression coefficients $\beta$ and noise variance $\sigma^2$.
+be a [linear regression model](/D/mlr) with measured $n \times 1$ data vector $y$, known $n \times p$ design matrix $X$, known $n \times n$ covariance structure $V$ as well as unknown $p \times 1$ regression coefficients $\beta$ and unknown noise variance $\sigma^2$.
 
 Then, the [conjugate prior](/D/prior-conj) for this model is a [normal-gamma distribution](/D/ng)
 
@@ -82,13 +82,13 @@ $$
 where $\tilde{X} = \left( X^\mathrm{T} P X \right)^{-1} X^\mathrm{T} P$ and $Q = \tilde{X}^\mathrm{T} \left( X^\mathrm{T} P X \right) \tilde{X}$.
 
 <br>
-In other words, the [likelihood function](/D/lf) is proportional to a power of $\tau$ times an exponential of $\tau$ and an exponential of a squared form of $\beta$, weighted by $\tau$:
+In other words, the [likelihood function](/D/lf) is proportional to a power of $\tau$, times an exponential of $\tau$ and an exponential of a squared form of $\beta$, weighted by $\tau$:
 
 $$ \label{eq:GLM-LF-s4}
 p(y|\beta,\tau) \propto \tau^{n/2} \cdot \exp\left[ -\frac{\tau}{2} \left( y^\mathrm{T} P y - y^\mathrm{T} Q y \right) \right] \cdot \exp\left[ -\frac{\tau}{2} (\beta - \tilde{X}y)^\mathrm{T} X^\mathrm{T} P X (\beta - \tilde{X}y) \right] \; .
 $$
 
-The same is true for a normal gamma distribution over $\beta$ and $\tau$
+The same is true for a [normal-gamma distribution](/D/ng) over $\beta$ and $\tau$
 
 $$ \label{eq:BLR-prior-s1}
 p(\beta,\tau) = \mathcal{N}(\beta; \mu_0, (\tau \Lambda_0)^{-1}) \cdot \mathrm{Gam}(\tau; a_0, b_0)

P/mblr-lme.md

@@ -30,28 +30,28 @@ $$
 be a [general linear model](/D/glm) with measured $n \times v$ data matrix $Y$, known $n \times p$ design matrix $X$, known $n \times n$ [covariance structure](/D/matn) $V$ as well as unknown $p \times v$ regression coefficients $B$ and unknown $v \times v$ [noise covariance](/D/matn) $\Sigma$. Moreover, assume a [normal-Wishart prior distribution](/P/mblr-prior) over the model parameters $B$ and $T = \Sigma^{-1}$:
 
 $$ \label{eq:GLM-NW-prior}
-p(B,T) = \mathcal{MN}(B; M_0, \Lambda_0^{-1}, T^{-1}) \cdot \mathcal{W}(T; P_0^{-1}, \nu_0) \; .
+p(B,T) = \mathcal{MN}(B; M_0, \Lambda_0^{-1}, T^{-1}) \cdot \mathcal{W}(T; \Omega_0^{-1}, \nu_0) \; .
 $$
 
 Then, the [log model evidence](/D/lme) for this model is
 
-\begin{equation} \label{eq:GLM-NW-LME}
+$$ \label{eq:GLM-NW-LME}
 \begin{split}
 \log p(y|m) = & \frac{v}{2} \log |P| - \frac{nv}{2} \log (2 \pi)  + \frac{v}{2} \log |\Lambda_0| - \frac{v}{2} \log |\Lambda_n| + \\
-& \frac{\nu_0}{2} \log\left| \frac{1}{2} P_0 \right| - \frac{\nu_n}{2} \log\left| \frac{1}{2} P_n \right| + \log \Gamma_v \left( \frac{\nu_n}{2} \right) - \log \Gamma_v \left( \frac{\nu_0}{2} \right)
+& \frac{\nu_0}{2} \log\left| \frac{1}{2} \Omega_0 \right| - \frac{\nu_n}{2} \log\left| \frac{1}{2} \Omega_n \right| + \log \Gamma_v \left( \frac{\nu_n}{2} \right) - \log \Gamma_v \left( \frac{\nu_0}{2} \right)
 \end{split}
-\end{equation}
+$$
 
 where the [posterior hyperparameters](/D/post) are given by
 
-\begin{equation} \label{eq:GLM-NW-post-par}
+$$ \label{eq:GLM-NW-post-par}
 \begin{split}
 M_n &= \Lambda_n^{-1} (X^\mathrm{T} P Y + \Lambda_0 M_0) \\
 \Lambda_n &= X^\mathrm{T} P X + \Lambda_0 \\
-P_n &= P_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\
+\Omega_n &= \Omega_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\
 \nu_n &= \nu_0 + n \; .
 \end{split}
-\end{equation}
+$$
 
 
 **Proof:** According to the [law of marginal probability](/D/prob-marg), the [model evidence](/D/ml) for this model is:
@@ -83,48 +83,48 @@ using the $v \times v$ [precision matrix](/D/precmat) $T = \Sigma^{-1}$ and the
 <br>
 When [deriving the posterior distribution](/P/mblr-post) $p(B,T|Y)$, the joint likelihood $p(Y,B,T)$ is obtained as
 
-\begin{equation} \label{eq:GLM-NW-LME-s1}
+$$ \label{eq:GLM-NW-LME-s1}
 \begin{split}
-p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( P_0 T \right) \right] \cdot \\
+p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\
 & \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (B-M_n)^\mathrm{T} \Lambda_n (B-M_n) + (Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n) \right] \right) \right] \; .
 \end{split}
-\end{equation}
+$$
 
 Using the [probability density function of the matrix-normal distribution](/P/matn-pdf), we can rewrite this as
 
-\begin{equation} \label{eq:GLM-NW-LME-s2}
+$$ \label{eq:GLM-NW-LME-s2}
 \begin{split}
-p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{(2 \pi)^{pv}}{|T|^p |\Lambda_n|^v}} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( P_0 T \right) \right] \cdot \\
+p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{(2 \pi)^{pv}}{|T|^p |\Lambda_n|^v}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\
 & \mathcal{MN}(B; M_n, \Lambda_n^{-1}, T^{-1}) \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \right] \right) \right] \; .
 \end{split}
-\end{equation}
+$$
 
 Now, $B$ can be integrated out easily:
 
-\begin{equation} \label{eq:GLM-NW-LME-s3}
+$$ \label{eq:GLM-NW-LME-s3}
 \begin{split}
-\int p(Y,B,T) \, \mathrm{d}B = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|\Lambda_0|^v}{|\Lambda_n|^v}} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \cdot \\
-& \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ P_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \right] \right) \right] \; .
+\int p(Y,B,T) \, \mathrm{d}B = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|\Lambda_0|^v}{|\Lambda_n|^v}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \cdot \\
+& \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ \Omega_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \right] \right) \right] \; .
 \end{split}
-\end{equation}
+$$
 
 Using the [probability density function of the Wishart distribution](/P/wish-pdf), we can rewrite this as
 
 $$ \label{eq:GLM-NW-LME-s4}
-\int p(Y,B,T) \, \mathrm{d}B = \sqrt{\frac{|P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|\Lambda_0|^v}{|\Lambda_n|^v}} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} \sqrt{\frac{2^{\nu_n v}}{|P_n|^{\nu_n}}} \, \frac{\Gamma_v \left( \frac{\nu_n}{2} \right)}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot \mathcal{W}(T; P_n^{-1}, \nu_n) \; .
+\int p(Y,B,T) \, \mathrm{d}B = \sqrt{\frac{|P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|\Lambda_0|^v}{|\Lambda_n|^v}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \sqrt{\frac{2^{\nu_n v}}{|\Omega_n|^{\nu_n}}} \, \frac{\Gamma_v \left( \frac{\nu_n}{2} \right)}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot \mathcal{W}(T; \Omega_n^{-1}, \nu_n) \; .
 $$
 
 Finally, $T$ can also be integrated out:
 
 $$ \label{eq:GLM-NW-LME-s5}
-\iint p(Y,B,T) \, \mathrm{d}B \, \mathrm{d}T = \sqrt{\frac{|P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|\Lambda_0|^v}{|\Lambda_n|^v}} \sqrt{\frac{\left| \frac{1}{2} P_0 \right|^{\nu_0}}{\left| \frac{1}{2} P_n \right|^{\nu_n}}} \, \frac{\Gamma_v \left( \frac{\nu_n}{2} \right)}{\Gamma_v \left( \frac{\nu_0}{2} \right)} = p(y|m) \; .
+\iint p(Y,B,T) \, \mathrm{d}B \, \mathrm{d}T = \sqrt{\frac{|P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|\Lambda_0|^v}{|\Lambda_n|^v}} \sqrt{\frac{\left| \frac{1}{2} \Omega_0 \right|^{\nu_0}}{\left| \frac{1}{2} \Omega_n \right|^{\nu_n}}} \, \frac{\Gamma_v \left( \frac{\nu_n}{2} \right)}{\Gamma_v \left( \frac{\nu_0}{2} \right)} = p(y|m) \; .
 $$
 
 Thus, the [log model evidence](/D/lme) of this model is given by
 
-\begin{equation} \label{eq:GLM-NW-LME-s6}
+$$ \label{eq:GLM-NW-LME-s6}
 \begin{split}
 \log p(y|m) = & \frac{v}{2} \log |P| - \frac{nv}{2} \log (2 \pi)  + \frac{v}{2} \log |\Lambda_0| - \frac{v}{2} \log |\Lambda_n| + \\
-& \frac{\nu_0}{2} \log\left| \frac{1}{2} P_0 \right| - \frac{\nu_n}{2} \log\left| \frac{1}{2} P_n \right| + \log \Gamma_v \left( \frac{\nu_n}{2} \right) - \log \Gamma_v \left( \frac{\nu_0}{2} \right) \; .
+& \frac{\nu_0}{2} \log\left| \frac{1}{2} \Omega_0 \right| - \frac{\nu_n}{2} \log\left| \frac{1}{2} \Omega_n \right| + \log \Gamma_v \left( \frac{\nu_n}{2} \right) - \log \Gamma_v \left( \frac{\nu_0}{2} \right) \; .
 \end{split}
-\end{equation}
+$$

P/mblr-post.md

@@ -36,13 +36,13 @@ $$
 be a [general linear model](/D/glm) with measured $n \times v$ data matrix $Y$, known $n \times p$ design matrix $X$, known $n \times n$ [covariance structure](/D/matn) $V$ as well as unknown $p \times v$ regression coefficients $B$ and unknown $v \times v$ [noise covariance](/D/matn) $\Sigma$. Moreover, assume a [normal-Wishart prior distribution](/P/mblr-prior) over the model parameters $B$ and $T = \Sigma^{-1}$:
 
 $$ \label{eq:GLM-NW-prior}
-p(B,T) = \mathcal{MN}(B; M_0, \Lambda_0^{-1}, T^{-1}) \cdot \mathcal{W}(T; P_0^{-1}, \nu_0) \; .
+p(B,T) = \mathcal{MN}(B; M_0, \Lambda_0^{-1}, T^{-1}) \cdot \mathcal{W}(T; \Omega_0^{-1}, \nu_0) \; .
 $$
 
 Then, the [posterior distribution](/D/post) is also a [normal-Wishart distribution](/D/nw)
 
 $$ \label{eq:GLM-NW-post}
-p(B,T|Y) = \mathcal{MN}(B; M_n, \Lambda_n^{-1}, T^{-1}) \cdot \mathcal{W}(T; P_n^{-1}, \nu_n)
+p(B,T|Y) = \mathcal{MN}(B; M_n, \Lambda_n^{-1}, T^{-1}) \cdot \mathcal{W}(T; \Omega_n^{-1}, \nu_n)
 $$
 
 and the [posterior hyperparameters](/D/post) are given by
@@ -51,7 +51,7 @@ $$ \label{eq:GLM-NW-post-par}
 \begin{split}
 M_n &= \Lambda_n^{-1} (X^\mathrm{T} P Y + \Lambda_0 M_0) \\
 \Lambda_n &= X^\mathrm{T} P X + \Lambda_0 \\
-P_n &= P_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\
+\Omega_n &= \Omega_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\
 \nu_n &= \nu_0 + n \; .
 \end{split}
 $$
@@ -91,15 +91,15 @@ $$ \label{eq:GLM-NW-JL-s1}
 p(Y,B,T) = \; & p(Y|B,T) \, p(B,T) \\
 = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \, \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T (Y-XB)^\mathrm{T} P (Y-XB) \right) \right] \cdot \\
 & \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \, \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T (B-M_0)^\mathrm{T} \Lambda_0 (B-M_0) \right) \right] \cdot \\
-& \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( P_0 T \right) \right] \; .
+& \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \; .
 \end{split}
 $$
 
 Collecting identical variables gives:
 
 $$ \label{eq:GLM-NW-JL-s2}
 \begin{split}
-p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( P_0 T \right) \right] \cdot \\
+p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\
 & \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (Y-XB)^\mathrm{T} P (Y-XB) + (B-M_0)^\mathrm{T} \Lambda_0 (B-M_0) \right] \right) \right] \; .
 \end{split}
 $$
@@ -108,7 +108,7 @@ Expanding the products in the exponent gives:
 
 $$ \label{eq:GLM-NW-JL-s3}
 \begin{split}
-p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( P_0 T \right) \right] \cdot \\
+p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\
 & \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \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. \right. \\
 & \hphantom{\exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ \right. \right. \right. \!\!\!} \; \left. \left. \left. B^\mathrm{T} \Lambda_0 B - B^\mathrm{T} \Lambda_0 M_0 - M_0^\mathrm{T} \Lambda_0 B + M_0^\mathrm{T} \Lambda_0 \mu_0 \right] \right) \right] \; .
 \end{split}
@@ -118,7 +118,7 @@ Completing the square over $B$, we finally have
 
 $$ \label{eq:GLM-NW-JL-s4}
 \begin{split}
-p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|P_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( P_0 T \right) \right] \cdot \\
+p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\
 & \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (B-M_n)^\mathrm{T} \Lambda_n (B-M_n) + (Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n) \right] \right) \right] \; .
 \end{split}
 $$
@@ -135,14 +135,14 @@ $$
 Ergo, the joint likelihood is proportional to
 
 $$ \label{eq:GLM-NW-JL-s5}
-p(Y,B,T) \propto |T|^{p/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (B-M_n)^\mathrm{T} \Lambda_n (B-M_n) \right] \right) \right] \cdot |T|^{(\nu_n-v-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( P_n T \right) \right]
+p(Y,B,T) \propto |T|^{p/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (B-M_n)^\mathrm{T} \Lambda_n (B-M_n) \right] \right) \right] \cdot |T|^{(\nu_n-v-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_n T \right) \right]
 $$
 
 with the [posterior hyperparameters](/D/post)
 
 $$ \label{eq:GLM-NW-post-T-par}
 \begin{split}
-P_n &= P_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\
+\Omega_n &= \Omega_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\
 \nu_n &= \nu_0 + n \; .
 \end{split}
 $$
@@ -156,7 +156,7 @@ $$
 From the remaining term, we can isolate the posterior distribution over $T$:
 
 $$ \label{eq:GLM-NW-post-T}
-p(T|Y) = \mathcal{W}(T; P_n^{-1}, \nu_n) \; .
+p(T|Y) = \mathcal{W}(T; \Omega_n^{-1}, \nu_n) \; .
 $$
 
 Together, \eqref{eq:GLM-NW-post-B} and \eqref{eq:GLM-NW-post-T} constitute the [joint](/D/prob-joint) [posterior distribution](/D/post) of $B$ and $T$.