update week 37

2024-09-09 06:10:45 +02:00
parent 0cc7d1ed49
commit b16b859d7d
8 changed files with 194 additions and 194 deletions
@@ -316,7 +316,7 @@ $$
 p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
 $$

-<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta} \)!   </p>
+<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta}) \)!   </p>

 <p>
 <!-- navigation buttons at the bottom of the page -->
@@ -302,11 +302,11 @@ unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear re
 \( \mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta} \), which means that the estimator of the regression parameters is unbiased.
 </p>

-<p>We found also that the variance of the estimate of the \( j \)-th regression coefficient is
+<p>In the exercises this week we show that the variance of the estimate of the \( j \)-th regression coefficient is
 \( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj}  \).
 </p>

-<p>This quantity will be used to
+<p>This quantity can be used to
 construct a confidence interval for the estimates.
 </p>

@@ -578,7 +578,7 @@ p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol
 $$
 <p>&nbsp;<br>

-<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta} \)!   </p>
+<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta}) \)!   </p>
 </section>

 <section>
@@ -998,11 +998,11 @@ unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear re
 \( \mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta} \), which means that the estimator of the regression parameters is unbiased.
 </p>

-<p>We found also that the variance of the estimate of the \( j \)-th regression coefficient is
+<p>In the exercises this week we show that the variance of the estimate of the \( j \)-th regression coefficient is
 \( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj}  \).
 </p>

-<p>This quantity will be used to
+<p>This quantity can be used to
 construct a confidence interval for the estimates.
 </p>
 </section>
@@ -577,7 +577,7 @@ $$
 p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
 $$

-<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta} \)!   </p>
+<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta}) \)!   </p>

 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="ridge-and-bayes">Ridge and Bayes </h2>
@@ -944,11 +944,11 @@ unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear re
 \( \mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta} \), which means that the estimator of the regression parameters is unbiased.
 </p>

-<p>We found also that the variance of the estimate of the \( j \)-th regression coefficient is
+<p>In the exercises this week we show that the variance of the estimate of the \( j \)-th regression coefficient is
 \( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj}  \).
 </p>

-<p>This quantity will be used to
+<p>This quantity can be used to
 construct a confidence interval for the estimates.
 </p>

@@ -654,7 +654,7 @@ $$
 p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
 $$

-<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta} \)!   </p>
+<p>We have a model for \( p(\boldsymbol{D}\vert\boldsymbol{\beta}) \) but need one for the <b>prior</b> \( p(\boldsymbol{\beta}) \)!   </p>

 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
 <h2 id="ridge-and-bayes">Ridge and Bayes </h2>
@@ -1021,11 +1021,11 @@ unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear re
 \( \mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta} \), which means that the estimator of the regression parameters is unbiased.
 </p>

-<p>We found also that the variance of the estimate of the \( j \)-th regression coefficient is
+<p>In the exercises this week we show that the variance of the estimate of the \( j \)-th regression coefficient is
 \( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj}  \).
 </p>

-<p>This quantity will be used to
+<p>This quantity can be used to
 construct a confidence interval for the estimates.
 </p>

@@ -339,7 +339,7 @@ p(\bm{\beta}\vert\bm{D})\propto p(\bm{D}\vert\bm{\beta})p(\bm{\beta}).
 \]
 !et

-We have a model for $p(\bm{D}\vert\bm{\beta})$ but need one for the _prior_ $p(\bm{\beta}$!   
+We have a model for $p(\bm{D}\vert\bm{\beta})$ but need one for the _prior_ $p(\bm{\beta})$!   


 !split
@@ -710,10 +710,10 @@ unknown parameter such as the parameters $\bm{\beta}$ from linear regression.
 With the OLS expressions for the parameters $\bm{\beta}$ we found 
 $\mathbb{E}(\bm{\beta}) = \bm{\beta}$, which means that the estimator of the regression parameters is unbiased.

-We found also that the variance of the estimate of the $j$-th regression coefficient is
+In the exercises this week we show that the variance of the estimate of the $j$-th regression coefficient is
 $\bm{\sigma}^2 (\bm{\beta}_j ) = \bm{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} $.

-This quantity will be used to
+This quantity can be used to
 construct a confidence interval for the estimates.