Merge pull request #68 from StatProofBook/master

JoramSoch · web-flow · commit 56121a08d522 · 2021-10-27T15:37:48.000+02:00
update to master
diff --git a/D/cfm.md b/D/cfm.md
@@ -5,9 +5,9 @@ mathjax: true
 author: "Joram Soch"
 affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
-date: 2021-10-21 17:01
+date: 2021-10-21 17:01:00
 
-title: "General linear model"
+title: "Corresponding forward model"
 chapter: "Statistical Models"
 section: "Multivariate normal data"
 topic: "Inverse general linear model"
diff --git a/D/iglm.md b/D/iglm.md
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
 date: 2021-10-21 15:31:00
 
-title: "General linear model"
+title: "Inverse general linear model"
 chapter: "Statistical Models"
 section: "Multivariate normal data"
 topic: "Inverse general linear model"
@@ -28,7 +28,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let there be a [general linear models](/D/glm) of measured data $Y \in \mathbb{R}^{n \times v}$ in terms of the [design matrix](/D/glm) $X \in \mathbb{R}^{n \times p}$:
+**Definition:** Let there be a [general linear model](/D/glm) of measured data $Y \in \mathbb{R}^{n \times v}$ in terms of the [design matrix](/D/glm) $X \in \mathbb{R}^{n \times p}$:
 
 $$ \label{eq:glm}
 Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma) \; .
diff --git a/D/tglm.md b/D/tglm.md
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
 date: 2021-10-21 14:43:00
 
-title: "General linear model"
+title: "Transformed general linear model"
 chapter: "Statistical Models"
 section: "Multivariate normal data"
 topic: "Transformed general linear model"
diff --git a/I/Definition_by_Author.md b/I/Definition_by_Author.md
@@ -4,7 +4,7 @@ title: "Definition by Author"
 ---
 
 
-### JoramSoch (154 definitions)
+### JoramSoch (157 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Alternative hypothesis](/D/h1)
@@ -28,6 +28,7 @@ title: "Definition by Author"
 - [Continuous uniform distribution](/D/cuni)
 - [Correlation](/D/corr)
 - [Correlation matrix](/D/corrmat)
+- [Corresponding forward model](/D/cfm)
 - [Covariance](/D/cov)
 - [Covariance matrix](/D/covmat)
 - [Critical value](/D/cval)
@@ -61,6 +62,7 @@ title: "Definition by Author"
 - [General linear model](/D/glm)
 - [Generative model](/D/gm)
 - [Informative and non-informative prior distribution](/D/prior-inf)
+- [Inverse general linear model](/D/iglm)
 - [Joint cumulative distribution function](/D/cdf-joint)
 - [Joint differential entropy](/D/dent-joint)
 - [Joint entropy](/D/ent-joint)
@@ -149,6 +151,7 @@ title: "Definition by Author"
 - [Statistical independence](/D/ind)
 - [Test statistic](/D/tstat)
 - [Total sum of squares](/D/tss)
+- [Transformed general linear model](/D/tglm)
 - [Uniform and non-uniform prior distribution](/D/prior-uni)
 - [Uniform-prior log model evidence](/D/uplme)
 - [Univariate Gaussian](/D/ug)
diff --git a/I/Definition_by_Number.md b/I/Definition_by_Number.md
@@ -166,3 +166,6 @@ title: "Definition by Number"
 | D157 | suni | [Standard uniform distribution](/D/suni) | JoramSoch | 2021-07-23 |
 | D158 | prob-ax | [Kolmogorov axioms of probability](/D/prob-ax) | JoramSoch | 2021-07-30 |
 | D159 | cf | [Characteristic function](/D/cf) | JoramSoch | 2021-09-22 |
+| D160 | tglm | [Transformed general linear model](/D/tglm) | JoramSoch | 2021-10-21 |
+| D161 | iglm | [Inverse general linear model](/D/iglm) | JoramSoch | 2021-10-21 |
+| D162 | cfm | [Corresponding forward model](/D/cfm) | JoramSoch | 2021-10-21 |
diff --git a/I/Definition_by_Topic.md b/I/Definition_by_Topic.md
@@ -36,6 +36,7 @@ title: "Definition by Topic"
 - [Continuous uniform distribution](/D/cuni)
 - [Correlation](/D/corr)
 - [Correlation matrix](/D/corrmat)
+- [Corresponding forward model](/D/cfm)
 - [Covariance](/D/cov)
 - [Covariance matrix](/D/covmat)
 - [Critical value](/D/cval)
@@ -86,6 +87,7 @@ title: "Definition by Topic"
 ### I
 
 - [Informative and non-informative prior distribution](/D/prior-inf)
+- [Inverse general linear model](/D/iglm)
 
 ### J
 
@@ -208,6 +210,7 @@ title: "Definition by Topic"
 
 - [Test statistic](/D/tstat)
 - [Total sum of squares](/D/tss)
+- [Transformed general linear model](/D/tglm)
 
 ### U
 
diff --git a/I/Proof_by_Author.md b/I/Proof_by_Author.md
@@ -4,7 +4,7 @@ title: "Proof by Author"
 ---
 
 
-### JoramSoch (254 proofs)
+### JoramSoch (260 proofs)
 
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
@@ -17,6 +17,7 @@ title: "Proof by Author"
 - [Bayes' rule](/P/bayes-rule)
 - [Bayes' theorem](/P/bayes-th)
 - [Bayesian model averaging in terms of log model evidences](/P/bma-lme)
+- [Best linear unbiased estimator for the inverse general linear model](/P/iglm-blue)
 - [Characteristic function of a function of a random variable](/P/cf-fct)
 - [Concavity of the Shannon entropy](/P/ent-conc)
 - [Conditional distributions of the multivariate normal distribution](/P/mvn-cond)
@@ -59,9 +60,13 @@ title: "Proof by Author"
 - [Differential entropy of the multivariate normal distribution](/P/mvn-dent)
 - [Differential entropy of the normal distribution](/P/norm-dent)
 - [Differential entropy of the normal-gamma distribution](/P/ng-dent)
+- [Distribution of the inverse general linear model](/P/iglm-dist)
+- [Distribution of the transformed general linear model](/P/tglm-dist)
 - [Distributional transformation using cumulative distribution function](/P/cdf-dt)
 - [Equivalence of matrix-normal distribution and multivariate normal distribution](/P/matn-mvn)
+- [Equivalence of parameter estimates from the transformed general linear model](/P/tglm-para)
 - [Exceedance probabilities for the Dirichlet distribution](/P/dir-ep)
+- [Existence of a corresponding forward model](/P/cfm-exist)
 - [Expectation of a quadratic form](/P/mean-qf)
 - [Expectation of the cross-validated log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-cvlbfmean)
 - [Expectation of the log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-lbfmean)
@@ -158,6 +163,7 @@ title: "Proof by Author"
 - [Ordinary least squares for the general linear model](/P/glm-ols)
 - [Paired t-test for dependent observations](/P/ug-ttestp)
 - [Paired z-test for dependent observations](/P/ugkv-ztestp)
+- [Parameters of the corresponding forward model](/P/cfm-para)
 - [Partition of a covariance matrix into expected values](/P/covmat-mean)
 - [Partition of covariance into expected values](/P/cov-mean)
 - [Partition of sums of squares in ordinary least squares](/P/mlr-pss)
diff --git a/I/Proof_by_Number.md b/I/Proof_by_Number.md
@@ -270,3 +270,9 @@ title: "Proof by Number"
 | P262 | dent-noninv | [Non-invariance of the differential entropy under change of variables](/P/dent-noninv) | JoramSoch | 2021-10-07 |
 | P263 | t-pdf | [Probability density function of the t-distribution](/P/t-pdf) | JoramSoch | 2021-10-12 |
 | P264 | f-pdf | [Probability density function of the F-distribution](/P/f-pdf) | JoramSoch | 2021-10-12 |
+| P265 | tglm-dist | [Distribution of the transformed general linear model](/P/tglm-dist) | JoramSoch | 2021-10-21 |
+| P266 | tglm-para | [Equivalence of parameter estimates from the transformed general linear model](/P/tglm-para) | JoramSoch | 2021-10-21 |
+| P267 | iglm-dist | [Distribution of the inverse general linear model](/P/iglm-dist) | JoramSoch | 2021-10-21 |
+| P268 | iglm-blue | [Best linear unbiased estimator for the inverse general linear model](/P/iglm-blue) | JoramSoch | 2021-10-21 |
+| P269 | cfm-para | [Parameters of the corresponding forward model](/P/cfm-para) | JoramSoch | 2021-10-21 |
+| P270 | cfm-exist | [Existence of a corresponding forward model](/P/cfm-exist) | JoramSoch | 2021-10-21 |
diff --git a/I/Proof_by_Topic.md b/I/Proof_by_Topic.md
@@ -23,6 +23,7 @@ title: "Proof by Topic"
 - [Bayes' rule](/P/bayes-rule)
 - [Bayes' theorem](/P/bayes-th)
 - [Bayesian model averaging in terms of log model evidences](/P/bma-lme)
+- [Best linear unbiased estimator for the inverse general linear model](/P/iglm-blue)
 
 ### C
 
@@ -72,13 +73,17 @@ title: "Proof by Topic"
 - [Differential entropy of the multivariate normal distribution](/P/mvn-dent)
 - [Differential entropy of the normal distribution](/P/norm-dent)
 - [Differential entropy of the normal-gamma distribution](/P/ng-dent)
+- [Distribution of the inverse general linear model](/P/iglm-dist)
+- [Distribution of the transformed general linear model](/P/tglm-dist)
 - [Distributional transformation using cumulative distribution function](/P/cdf-dt)
 
 ### E
 
 - [Encompassing Prior Method for computing Bayes Factors](/P/bf-ep)
 - [Equivalence of matrix-normal distribution and multivariate normal distribution](/P/matn-mvn)
+- [Equivalence of parameter estimates from the transformed general linear model](/P/tglm-para)
 - [Exceedance probabilities for the Dirichlet distribution](/P/dir-ep)
+- [Existence of a corresponding forward model](/P/cfm-exist)
 - [Expectation of a quadratic form](/P/mean-qf)
 - [Expectation of the cross-validated log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-cvlbfmean)
 - [Expectation of the log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-lbfmean)
@@ -208,6 +213,7 @@ title: "Proof by Topic"
 
 - [Paired t-test for dependent observations](/P/ug-ttestp)
 - [Paired z-test for dependent observations](/P/ugkv-ztestp)
+- [Parameters of the corresponding forward model](/P/cfm-para)
 - [Partition of a covariance matrix into expected values](/P/covmat-mean)
 - [Partition of covariance into expected values](/P/cov-mean)
 - [Partition of sums of squares in ordinary least squares](/P/mlr-pss)
diff --git a/P/cfm-exist.md b/P/cfm-exist.md
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
 date: 2021-10-21 17:43:00
 
-title: "Existence of the corresponding forward model"
+title: "Existence of a corresponding forward model"
 chapter: "Statistical Models"
 section: "Multivariate normal data"
 topic: "Inverse general linear model"
@@ -28,7 +28,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
+**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W = f(Y,X) \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
 
 $$ \label{eq:bda}
 \hat{X} = Y W \; .
diff --git a/P/cfm-para.md b/P/cfm-para.md
@@ -28,7 +28,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
+**Theorem:** Let there be observations $Y \in \mathbb{R}^{n \times v}$ and $X \in \mathbb{R}^{n \times p}$ and consider a weight matrix $W = f(Y,X) \in \mathbb{R}^{v \times p}$ predicting $X$ from $Y$:
 
 $$ \label{eq:bda}
 \hat{X} = Y W \; .
@@ -40,7 +40,7 @@ $$ \label{eq:cfm-para}
 A = \Sigma_y W \Sigma_x^{-1}
 $$
 
-with the [sample covariance](/D/cov-samp)
+with the "[sample covariances](/D/cov-samp)"
 
 $$ \label{eq:Sx-Sy}
 \begin{split}
diff --git a/P/iglm-blue.md b/P/iglm-blue.md
@@ -73,7 +73,13 @@ $$ \label{eq:W-hat-dist}
 \tilde{W} = M X \sim \mathcal{MN}(M Y W, M V M^T, \Sigma_x) \;
 $$
 
-which requires that $M Y = I_v$. This is fulfilled by any matrix $M = (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D$ where $D$ is a $v \times n$ matrix which satisfies $D Y = 0$.
+[which requires](/P/matn-mean) that $M Y = I_v$. This is fulfilled by any matrix
+
+$$ \label{eq:M-D}
+M = (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D
+$$
+
+where $D$ is a $v \times n$ matrix which satisfies $D Y = 0$.
 
 <br>
 3) Third, the [best linear unbiased estimator](/D/blue) is the one with minimum [variance](/D/var), i.e. the one that minimizes the expected Frobenius norm
@@ -85,26 +91,32 @@ $$
 Using the [matrix-normal distribution](/D/matn) of $\tilde{W}$ from \eqref{eq:W-hat-dist}
 
 $$ \label{eq:W-hat-W-dist}
-\left( \tilde{W} - W \right) \sim \mathcal{MN}(0, M V M^T, \Sigma_x)
+\left( \tilde{W} - W \right) \sim \mathcal{MN}(0, M V M^\mathrm{T}, \Sigma_x)
 $$
 
 and the property of the [Wishart distribution](/D/wish)
 
 $$ \label{eq:E-XX}
-X \sim \mathcal{MN}(0, U, V) \quad \Rightarrow \quad \left\langle X X^T \right\rangle = \mathrm{tr}(V) \, U \; ,
+X \sim \mathcal{MN}(0, U, V) \quad \Rightarrow \quad \left\langle X X^\mathrm{T} \right\rangle = \mathrm{tr}(V) \, U \; ,
 $$
 
 this [variance](/D/var) can be evaluated as a function of $M$:
 
 $$ \label{eq:Var-M}
-\mathrm{Var}\left[ \tilde{W}(M) \right] = \mathrm{tr}(\Sigma_x) \; \mathrm{tr}(M V M^T) \; .
+\begin{split}
+\mathrm{Var}\left[ \tilde{W}(M) \right] &\overset{\eqref{eq:Var-W}}{=} \left\langle \mathrm{tr}\left[ (\tilde{W} - W)^\mathrm{T} (\tilde{W} - W) \right] \right\rangle \\
+&= \left\langle \mathrm{tr}\left[ (\tilde{W} - W) (\tilde{W} - W)^\mathrm{T} \right] \right\rangle \\
+&= \mathrm{tr}\left[ \left\langle (\tilde{W} - W) (\tilde{W} - W)^\mathrm{T} \right\rangle \right] \\
+&\overset{\eqref{eq:E-XX}}{=} \mathrm{tr}\left[ \mathrm{tr}(\Sigma_x) \, M V M^\mathrm{T} \right] \\
+&= \mathrm{tr}(\Sigma_x) \; \mathrm{tr}(M V M^\mathrm{T}) \; .
+\end{split}
 $$
 
 As a function of $D$ and using $D Y = 0$, it becomes:
 
 $$ \label{eq:Var-D}
 \begin{split}
-\mathrm{Var}\left[ \tilde{W}(D) \right] &= \mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[ \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right) V \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right)^\mathrm{T} \right] \\
+\mathrm{Var}\left[ \tilde{W}(D) \right] &\overset{\eqref{eq:M-D}}{=} \mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[ \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right) V \left( (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} + D \right)^\mathrm{T} \right] \\
 &= \mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[ (Y^\mathrm{T} V^{-1} Y)^{-1} \, Y^\mathrm{T} V^{-1} V V^{-1} Y \; (Y^\mathrm{T} V^{-1} Y)^{-1} + \right. \\
 &\hphantom{=\mathrm{tr}(\Sigma_x) \; \mathrm{tr}\!\left[\right.} \left. \, (Y^\mathrm{T} V^{-1} Y)^{-1} Y^\mathrm{T} V^{-1} V D^\mathrm{T} + D V V^{-1} Y (Y^\mathrm{T} V^{-1} Y)^{-1} + D V D^\mathrm{T} \right] \\
 &= \mathrm{tr}(\Sigma_x) \left[ \mathrm{tr}\!\left( (Y^\mathrm{T} V^{-1} Y)^{-1} \right) + \mathrm{tr}\!\left( D V D^\mathrm{T} \right) \right] \; .
diff --git a/P/iglm-dist.md b/P/iglm-dist.md
@@ -40,7 +40,7 @@ $$ \label{eq:iglm}
 X = Y W + N, \; N \sim \mathcal{MN}(0, V, \Sigma_x)
 $$
 
-where $W \in \mathbb{R}^{v \times p}$ is a matrix, such that $B \, W = I_p$, and the covariance across columns is $\Sigma_x = W^\mathrm{T} \Sigma W$.
+where $W \in \mathbb{R}^{v \times p}$ is a matrix, such that $B \, W = I_p$, and the [covariance across columns](/D/matn) is $\Sigma_x = W^\mathrm{T} \Sigma W$.
 
 
 **Proof:** The [linear transformation theorem for the matrix-normal distribution](/P/matn-ltt) states:
@@ -49,13 +49,13 @@ $$ \label{eq:matn-ltt}
 X \sim \mathcal{MN}(M, U, V) \quad \Rightarrow \quad Y = AXB + C \sim \mathcal{MN}(AMB+C, AUA^\mathrm{T}, B^\mathrm{T}VB) \; .
 $$
 
-The matrix $W$ exists, if the rows of $B \in \mathbb{R}^{p \times v}$ are linearly independent, such that $\mathrm{rk}(B) = p$. Then, right-multiplying the model \eqref{eq:glm} and applying \eqref{eq:matn-ltt} with $W$ yields
+The matrix $W$ exists, if the rows of $B \in \mathbb{R}^{p \times v}$ are linearly independent, such that $\mathrm{rk}(B) = p$. Then, right-multiplying the model \eqref{eq:glm} with $W$ and applying \eqref{eq:matn-ltt} yields
 
 $$ \label{eq:iglm-s1}
 Y W = X B W + E W, \; E W \sim \mathcal{MN}(0, V, W^\mathrm{T} \Sigma W) \; .
 $$
 
-Applying $B \, W = I_p$ and rearranging, we have
+Employing $B \, W = I_p$ and rearranging, we have
 
 $$ \label{eq:iglm-s2}
 X = Y W - E W, \; E W \sim \mathcal{MN}(0, V, W^\mathrm{T} \Sigma W) \; .
diff --git a/P/tglm-dist.md b/P/tglm-dist.md
@@ -50,7 +50,7 @@ $$ \label{eq:tglm}
 \hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}(0, U, \Sigma)
 $$
 
-where the covariance across rows is $U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1}$.
+where the [covariance across rows](/D/matn) is $U = ( X_t^\mathrm{T} V^{-1} X_t )^{-1}$.
 
 
 **Proof:** The [linear transformation theorem for the matrix-normal distribution](/P/matn-ltt) states:
@@ -75,13 +75,13 @@ Combining \eqref{eq:glm2-wls} with \eqref{eq:Y-dist}, the distribution of $\hat{
 
 $$ \label{eq:G-dist}
 \begin{split}
-\hat{\Gamma} &\sim \mathrm{MN}\left( \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] X B, \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] V \left[ V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1} \right], \Sigma \right) \\
-&\sim \mathrm{MN}\left( ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t \, T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \\
-&\sim \mathrm{MN}\left( T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \; .
+\hat{\Gamma} &\sim \mathcal{MN}\left( \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] X B, \left[ ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} \right] V \left[ V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1} \right], \Sigma \right) \\
+&\sim \mathcal{MN}\left( ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t \, T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} X_t ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \\
+&\sim \mathcal{MN}\left( T B, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right) \; .
 \end{split}
 $$
 
-This can also written as
+This can also be written as
 
 $$ \label{eq:tglm-qed}
 \hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}\left( 0, ( X_t^\mathrm{T} V^{-1} X_t )^{-1}, \Sigma \right)
diff --git a/P/tglm-para.md b/P/tglm-para.md
@@ -40,7 +40,7 @@ $$ \label{eq:tglm}
 \hat{\Gamma} = T B + H, \; H \sim \mathcal{MN}(0, U, \Sigma)
 $$
 
-which are linked to each other via
+which [are linked to each other](/P/tglm-dist) via
 
 $$ \label{eq:glm2-wls}
 \hat{\Gamma} = ( X_t^\mathrm{T} V^{-1} X_t )^{-1} X_t^\mathrm{T} V^{-1} Y
@@ -52,7 +52,7 @@ $$ \label{eq:X-Xt-T}
 X = X_t \, T \; .
 $$
 
-Then, the parameter estimates from \eqref{eq:glm1} and \eqref{eq:tglm} are equivalent.
+Then, the parameter estimates for $B$ from \eqref{eq:glm1} and \eqref{eq:tglm} are equivalent.
 
 
 **Proof:** The [weighted least squares parameter estimates](/P/glm-wls) for \eqref{eq:glm1} are given by
