From 80e0953a34b31b08b8dae557463832dc4b0e5c08 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 10 Sep 2021 08:05:53 +0200 Subject: [PATCH] cleaning typos --- doc/pub/week36/html/week36-reveal.html | 12 ++++++------ doc/pub/week36/html/week36-solarized.html | 12 ++++++------ doc/pub/week36/html/week36.html | 12 ++++++------ doc/pub/week36/ipynb/ipynb-week36-src.tar.gz | Bin 192 -> 192 bytes doc/pub/week36/ipynb/week36.ipynb | 12 ++++++------ doc/src/week36/week36.do.txt | 12 ++++++------ 6 files changed, 30 insertions(+), 30 deletions(-) diff --git a/doc/pub/week36/html/week36-reveal.html b/doc/pub/week36/html/week36-reveal.html index 3e9398634..2c3f35cd8 100644 --- a/doc/pub/week36/html/week36-reveal.html +++ b/doc/pub/week36/html/week36-reveal.html @@ -1870,7 +1870,7 @@ We can, based on our discussions of the variance of \( \boldsymbol{\beta} \) and

 
$$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{-(\frac{\beta_j^2}{2\tau^2}}. +p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. $$

 
@@ -1878,7 +1878,7 @@ $$ Our posterior probability becomes then (omitting the normalization factor which is just a constant)

 
$$ -p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{-(\frac{\beta_j^2}{2\tau^2}}. +p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. $$

 
@@ -1890,7 +1890,7 @@ constants terms that do not depend on \( \beta \), we have

 
$$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, +C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, $$

 
@@ -1914,7 +1914,7 @@ To derive the Lasso cost function, we simply replace the Gaussian prior with an

 
$$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{-(\frac{\vert\beta_j\vert}{\tau}}. +p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. $$

 
@@ -1922,7 +1922,7 @@ $$ Our posterior probability becomes then (omitting the normalization factor which is just a constant)

 
$$ -p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{-(\frac{\vert\beta_j\vert}{\tau}}. +p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. $$

 
@@ -1933,7 +1933,7 @@ constants terms that do not depend on \( \beta \), we have

 
$$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, +C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, $$

 
diff --git a/doc/pub/week36/html/week36-solarized.html b/doc/pub/week36/html/week36-solarized.html index f024c75c0..7bdfd2648 100644 --- a/doc/pub/week36/html/week36-solarized.html +++ b/doc/pub/week36/html/week36-solarized.html @@ -1852,13 +1852,13 @@ additional models for the prior. We can, based on our discussions of the variance of \( \boldsymbol{\beta} \) and the mean value, assume that the prior for the values \( \boldsymbol{\beta} \) is given by a Gaussian with mean value zero and variance \( \tau^2 \), that is $$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{-(\frac{\beta_j^2}{2\tau^2}}. +p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. $$

Our posterior probability becomes then (omitting the normalization factor which is just a constant) $$ -p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{-(\frac{\beta_j^2}{2\tau^2}}. +p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. $$

@@ -1868,7 +1868,7 @@ logarithm of the posterior probability. Doing so and leaving out the constants terms that do not depend on \( \beta \), we have $$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, +C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, $$ and replacing \( 1/2\tau^2 \) with \( \lambda \) we have @@ -1888,13 +1888,13 @@ which is our Ridge cost function! Nice, isn't it? To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution (Laplace in this case) with zero mean value, that is $$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{-(\frac{\vert\beta_j\vert}{\tau}}. +p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. $$

Our posterior probability becomes then (omitting the normalization factor which is just a constant) $$ -p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{-(\frac{\vert\beta_j\vert}{\tau}}. +p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. $$

@@ -1903,7 +1903,7 @@ logarithm of the posterior probability and leaving out the constants terms that do not depend on \( \beta \), we have $$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, +C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, $$ and replacing \( 1/\tau \) with \( \lambda \) we have diff --git a/doc/pub/week36/html/week36.html b/doc/pub/week36/html/week36.html index 216bca650..9255aba80 100644 --- a/doc/pub/week36/html/week36.html +++ b/doc/pub/week36/html/week36.html @@ -1857,13 +1857,13 @@ additional models for the prior. We can, based on our discussions of the variance of \( \boldsymbol{\beta} \) and the mean value, assume that the prior for the values \( \boldsymbol{\beta} \) is given by a Gaussian with mean value zero and variance \( \tau^2 \), that is $$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{-(\frac{\beta_j^2}{2\tau^2}}. +p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. $$

Our posterior probability becomes then (omitting the normalization factor which is just a constant) $$ -p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{-(\frac{\beta_j^2}{2\tau^2}}. +p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}. $$

@@ -1873,7 +1873,7 @@ logarithm of the posterior probability. Doing so and leaving out the constants terms that do not depend on \( \beta \), we have $$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, +C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2, $$ and replacing \( 1/2\tau^2 \) with \( \lambda \) we have @@ -1893,13 +1893,13 @@ which is our Ridge cost function! Nice, isn't it? To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution (Laplace in this case) with zero mean value, that is $$ -p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{-(\frac{\vert\beta_j\vert}{\tau}}. +p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. $$

Our posterior probability becomes then (omitting the normalization factor which is just a constant) $$ -p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{-(\frac{\vert\beta_j\vert}{\tau}}. +p(\boldsymbol{\beta}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}. $$

@@ -1908,7 +1908,7 @@ logarithm of the posterior probability and leaving out the constants terms that do not depend on \( \beta \), we have $$ -C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, +C(\boldsymbol{\beta}=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1, $$ and replacing \( 1/\tau \) with \( \lambda \) we have diff --git a/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz b/doc/pub/week36/ipynb/ipynb-week36-src.tar.gz index 74c7418d7ced8c77e17772cf43ce9427c6dd50e3..c23b9d50dceea8bd9ce3b37f50b14a9f5002fbdb 100644 GIT binary patch literal 192 zcmV;x06+g9iwFR?^*Uhy1MSbv3c@f92k@Qu6nTQtx~@A5?%+WX@dY}TxjNU*wnO*! z?gR9sco`z}cli?%LUPE~n_U*Uy9*XW2;q#um?=-DB&)feP?`W`l=389@st3eaT2ot z$Z{vWbk+$ooYGWhR2J2{xnZmuHia9~aqe<6-E?fvAULd8ZjkQTlQna_X z573q3rihSl^E1pa%p5kW^=^~+yN?!w5aMYHV`iL9iO5_}FlK;Lj5*5)B^(f@jFJe@ zawolX&I>!7(o|=uoz(B