Merge pull request #74 from StatProofBook/master

update to master

Author: JoramSoch

I/PbA.md

@@ -4,7 +4,7 @@ title: "Proof by Author"
 ---
 
 
-### JoramSoch (286 proofs)
+### JoramSoch (288 proofs)
 
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
@@ -74,6 +74,7 @@ title: "Proof by Author"
 - [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)
 - [Expected value of a non-negative random variable](/P/mean-nnrvar)
+- [Expected value of the trace of a matrix](/P/mean-tr)
 - [Expected value of x times ln(x) for a gamma distribution](/P/gam-xlogx)
 - [Exponential distribution is a special case of gamma distribution](/P/exp-gam)
 - [Expression of the cumulative distribution function of the normal distribution without the error function](/P/norm-cdfwerf)
@@ -267,6 +268,7 @@ title: "Proof by Author"
 - [Relationship between residual variance and sample variance in simple linear regression](/P/slr-resvar)
 - [Relationship between second raw moment, variance and mean](/P/momraw-2nd)
 - [Relationship between signal-to-noise ratio and R²](/P/snr-rsq)
+- [Sampling from the matrix-normal distribution](/P/matn-samp)
 - [Scaling of the variance upon multiplication with a constant](/P/var-scal)
 - [Second central moment is variance](/P/momcent-2nd)
 - [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr)

I/PbN.md

@@ -302,3 +302,5 @@ title: "Proof by Number"
 | P294 | dir-kl | [Kullback-Leibler divergence for the Dirichlet distribution](/P/dir-kl) | JoramSoch | 2021-12-02 |
 | P295 | wish-kl | [Kullback-Leibler divergence for the Wishart distribution](/P/wish-kl) | JoramSoch | 2021-12-02 |
 | P296 | matn-kl | [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl) | JoramSoch | 2021-12-02 |
+| P297 | matn-samp | [Sampling from the matrix-normal distribution](/P/matn-samp) | JoramSoch | 2021-12-07 |
+| P298 | mean-tr | [Expected value of the trace of a matrix](/P/mean-tr) | JoramSoch | 2021-12-07 |

I/PbT.md

@@ -87,6 +87,7 @@ title: "Proof by Topic"
 - [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)
 - [Expected value of a non-negative random variable](/P/mean-nnrvar)
+- [Expected value of the trace of a matrix](/P/mean-tr)
 - [Expected value of x times ln(x) for a gamma distribution](/P/gam-xlogx)
 - [Exponential distribution is a special case of gamma distribution](/P/exp-gam)
 - [Expression of the cumulative distribution function of the normal distribution without the error function](/P/norm-cdfwerf)
@@ -325,6 +326,7 @@ title: "Proof by Topic"
 
 ### S
 
+- [Sampling from the matrix-normal distribution](/P/matn-samp)
 - [Savage-Dickey Density Ratio for computing Bayes Factors](/P/bf-sddr)
 - [Scaling of the variance upon multiplication with a constant](/P/var-scal)
 - [Second central moment is variance](/P/momcent-2nd)

I/ToC.md

@@ -452,7 +452,7 @@ title: "Table of Contents"
    &emsp;&ensp; 5.1.4. **[Kullback-Leibler divergence](/P/matn-kl)** <br>
    &emsp;&ensp; 5.1.5. **[Linear transformation](/P/matn-ltt)** <br>
    &emsp;&ensp; 5.1.6. **[Transposition](/P/matn-trans)** <br>
-   &emsp;&ensp; 5.1.7. **[Drawing samples](/P/matn-ltt)** <br>
+   &emsp;&ensp; 5.1.7. **[Drawing samples](/P/matn-samp)** <br>
 
    5.2. Wishart distribution <br>
    &emsp;&ensp; 5.2.1. *[Definition](/D/wish)* <br>

P/matn-samp.md

@@ -11,7 +11,7 @@ title: "Sampling from the matrix-normal distribution"
 chapter: "Probability Distributions"
 section: "Matrix-variate continuous distributions"
 topic: "Matrix-normal distribution"
-theorem: "Linear transformation"
+theorem: "Drawing samples"
 
 sources:
   - authors: "Wikipedia"
@@ -67,4 +67,4 @@ Y = M + AXB &\sim \mathcal{MN}\left(M + A 0_{np} B, A I_n A^\mathrm{T}, B^\mathr
 \end{split}
 $$
 
-Thus, given $X$ defined by \eqref{eq:xij-dist}, $Y$ defined by \eqref{eq:matn-samp} is a [sample](/D/dist) from $\mathcal{N}\left(M, U, V \right)$.
+Thus, given $X$ defined by \eqref{eq:xij-dist}, $Y$ defined by \eqref{eq:matn-samp} is a [sample](/D/samp) from $\mathcal{N}\left(M, U, V \right)$.

P/mean-qf.md

@@ -63,7 +63,7 @@ $$ \label{eq:mean-qf-s2}
 \mathrm{E}\left[ X^\mathrm{T} A X \right] =  \mathrm{E}\left[ \mathrm{tr} \left( A X X^\mathrm{T} \right) \right] \; .
 $$
 
-Because [mean and trace are linear operators](/P/mean-lin), we have
+Because [mean and trace are linear operators](/P/mean-tr), we have
 
 $$ \label{eq:mean-qf-s3}
 \mathrm{E}\left[ X^\mathrm{T} A X \right] =  \mathrm{tr} \left( A \; \mathrm{E}\left[ X X^\mathrm{T} \right] \right) \; .

P/mean-tr.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $A$ be an $n \times n$ [random matrix](/D/rmat). Then, the [expected value](/D/mean) of the trace of $A$ is equal to the trace of the [expectation](/D/mean) of $A$:
+**Theorem:** Let $A$ be an $n \times n$ [random matrix](/D/rmat). Then, the [expectation](/D/mean) of the trace of $A$ is equal to the trace of the [expectation](/D/mean) of $A$:
 
 $$ \label{eq:mean-tr}
 \mathrm{E}\left[ \mathrm{tr}(A) \right] = \mathrm{tr}\left( \mathrm{E}[A] \right) \; .

P/mvn-kl.md

@@ -76,7 +76,7 @@ $$ \label{eq:mvn-KL-s3}
 \end{split}
 $$
 
-Because trace function and [expected value](/D/mean) are both linear operators, the expectation can be moved inside the trace:
+Because [trace function and expected value are both linear operators](/P/mean-tr), the expectation can be moved inside the trace:
 
 $$ \label{eq:mvn-KL-s4}
 \begin{split}