Merge pull request #156 from StatProofBook/master

update to master

Author: JoramSoch

I/DbA.md

@@ -13,7 +13,7 @@ title: "Definition by Author"
 
 - [Sample covariance](/D/cov-samp)
 
-### JoramSoch (191 definitions)
+### JoramSoch (192 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Alternative hypothesis](/D/h1)
@@ -65,6 +65,7 @@ title: "Definition by Author"
 - [Empirical Bayes](/D/eb)
 - [Empirical Bayes prior distribution](/D/prior-eb)
 - [Empirical Bayesian log model evidence](/D/eblme)
+- [Estimand, estimator and estimate](/D/est)
 - [Estimation matrix](/D/emat)
 - [Event space](/D/eve-spc)
 - [Exceedance probability](/D/prob-exc)

I/DbN.md

@@ -214,3 +214,4 @@ title: "Definition by Number"
 | D205 | stat | [Statistic](/D/stat) | JoramSoch | 2024-10-04 |
 | D206 | dof | [Degrees of freedom](/D/dof) | JoramSoch | 2024-10-11 |
 | D207 | dist-uni | [Unimodal and multimodal probability distribution](/D/dist-uni) | JoramSoch | 2024-10-25 |
+| D208 | est | [Estimand, estimator and estimate](/D/est) | JoramSoch | 2024-11-01 |

I/DbT.md

@@ -73,6 +73,7 @@ title: "Definition by Topic"
 - [Empirical Bayes prior distribution](/D/prior-eb)
 - [Empirical Bayesian log model evidence](/D/eblme)
 - [Encompassing model](/D/encm)
+- [Estimand, estimator and estimate](/D/est)
 - [Estimation matrix](/D/emat)
 - [Event space](/D/eve-spc)
 - [ex-Gaussian distribution](/D/exg)

I/PbA.md

@@ -12,7 +12,7 @@ title: "Proof by Author"
 
 - [Posterior predictive distribution is a marginal distribution of the joint likelihood](/P/postpred-jl)
 
-### JoramSoch (439 proofs)
+### JoramSoch (441 proofs)
 
 - [Accuracy and complexity for Bayesian linear regression](/P/blr-anc)
 - [Accuracy and complexity for Bayesian linear regression with known covariance](/P/blrkc-anc)
@@ -260,6 +260,8 @@ title: "Proof by Author"
 - [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)
 - [Mutual information of dependent and independent variables in the general linear model](/P/glm-mi)
+- [Mutual information of the bivariate normal distribution](/P/bvn-mi)
+- [Mutual information of the multivariate normal distribution](/P/mvn-mi)
 - [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

@@ -481,3 +481,5 @@ title: "Proof by Number"
 | P473 | norm-corrind | [Normally distributed and uncorrelated does not imply independent](/P/norm-corrind) | JoramSoch | 2024-10-04 |
 | P474 | norm-margjoint | [Marginally normal does not imply jointly normal](/P/norm-margjoint) | JoramSoch | 2024-10-11 |
 | P475 | bvn-lincomb | [Linear combination of bivariate normal random variables](/P/bvn-lincomb) | JoramSoch | 2024-10-25 |
+| P476 | bvn-mi | [Mutual information of the bivariate normal distribution](/P/bvn-mi) | JoramSoch | 2024-11-01 |
+| P477 | mvn-mi | [Mutual information of the multivariate normal distribution](/P/mvn-mi) | JoramSoch | 2024-11-01 |

I/PbT.md

@@ -302,6 +302,8 @@ title: "Proof by Topic"
 - [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)
 - [Mutual information of dependent and independent variables in the general linear model](/P/glm-mi)
+- [Mutual information of the bivariate normal distribution](/P/bvn-mi)
+- [Mutual information of the multivariate normal distribution](/P/mvn-mi)
 
 ### N
 

P/bvn-mi.md

@@ -37,7 +37,7 @@ $$
 Then, the [mutual information](/D/mi) of $X$ and $Y$ is
 
 $$ \label{eq:bvn-lincomb}
-\mathrm{I}(X,Y) = -\frac{1}{2} \ln (1-\rho^2)
+\mathrm{I}(X,Y) = -\frac{1}{2} \ln \left( 1-\rho^2 \right)
 $$
 
 where $\rho$ is the [correlation](/D/corr) of $X$ and $Y$.
@@ -49,7 +49,7 @@ $$ \label{eq:cmi-mjde}
 \mathrm{I}(X,Y) = \mathrm{h}(X) + \mathrm{h}(Y) - \mathrm{h}(X,Y) \; .
 $$
 
-The [marginal distributions of the multivariate normal distribution are also multivariate normal]
+The [marginal distributions of the multivariate normal distribution are also multivariate normal](/P/mvn-marg)
 
 $$ \label{eq:mvn-marg}
 \left[ \begin{matrix} X_1 \\ X_2 \end{matrix} \right] \sim
@@ -104,6 +104,6 @@ $$ \label{eq:bvn-mi}
 &= \frac{1}{2} \ln \left[ \frac{\sigma_1^2 \sigma_2^2}{\sigma_1^2 \sigma_2^2 - (\rho \, \sigma_1 \sigma_2)^2} \right] \\
 &= \frac{1}{2} \ln \left[ \frac{\sigma_1^2 \sigma_2^2}{\sigma_1^2 \sigma_2^2 (1-\rho^2)} \right] \\
 &= \frac{1}{2} \ln \left[ \frac{1}{1-\rho^2} \right] \\
-&= -\frac{1}{2} \ln (1-\rho^2) \; .
+&= -\frac{1}{2} \ln \left( 1-\rho^2 \right) \; .
 \end{split}
 $$

P/mvn-mi.md

@@ -49,7 +49,7 @@ $$ \label{eq:cmi-mjde}
 \mathrm{I}(X,Y) = \mathrm{h}(X) + \mathrm{h}(Y) - \mathrm{h}(X,Y) \; .
 $$
 
-The [marginal distributions of the multivariate normal distribution are also multivariate normal]
+The [marginal distributions of the multivariate normal distribution are also multivariate normal](/P/mvn-marg)
 
 $$ \label{eq:mvn-marg}
 \left[ \begin{matrix} X_1 \\ X_2 \end{matrix} \right] \sim