Morten Hjorth-Jensen
353 lines
19 KiB
HTML
353 lines
19 KiB
HTML
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<a class="navbar-brand" href="week36-bs.html">Week 36: Statistical interpretation of Linear Regression and Resampling techniques</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week36-bs001.html#plans-for-week-36" style="font-size: 80%;">Plans for week 36</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs002.html#thursday-september-9" style="font-size: 80%;">Thursday September 9</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs003.html#summary-from-last-week-and-examples" style="font-size: 80%;">Summary from last Week and Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs004.html#linear-regression-and-the-svd" style="font-size: 80%;">Linear Regression and the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs005.html#what-does-it-mean" style="font-size: 80%;">What does it mean?</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs006.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;">And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#code-for-svd-and-inversion-of-matrices" style="font-size: 80%;">Code for SVD and Inversion of Matrices</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs008.html#inverse-of-rectangular-matrix" style="font-size: 80%;">Inverse of Rectangular Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs009.html#ridge-and-lasso-regression" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs010.html#from-ols-to-ridge-and-lasso" style="font-size: 80%;">From OLS to Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs011.html#deriving-the-ridge-regression-equations" style="font-size: 80%;">Deriving the Ridge Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#svd-analysis" style="font-size: 80%;">SVD analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs013.html#interpreting-the-ridge-results" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs014.html#more-interpretations" style="font-size: 80%;">More interpretations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs015.html#deriving-the-lasso-regression-equations" style="font-size: 80%;">Deriving the Lasso Regression Equations</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs016.html#simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression" style="font-size: 80%;">Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs017.html#ridge-regression" style="font-size: 80%;">Ridge Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs018.html#lasso-regression" style="font-size: 80%;">Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs019.html#yet-another-example" style="font-size: 80%;">Yet another Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs020.html#the-ols-case" style="font-size: 80%;">The OLS case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs021.html#the-ridge-case" style="font-size: 80%;">The Ridge case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.html#writing-the-cost-function" style="font-size: 80%;">Writing the Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs023.html#lasso-case" style="font-size: 80%;">Lasso case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs024.html#the-first-case" style="font-size: 80%;">The first Case</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs025.html#simple-code-for-solving-the-above-problem" style="font-size: 80%;">Simple code for solving the above problem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs026.html#with-lasso-regression" style="font-size: 80%;">With Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs027.html#another-example-now-with-a-polynomial-fit" style="font-size: 80%;">Another Example, now with a polynomial fit</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs028.html#using-cvxopt" style="font-size: 80%;">Using CVXOPT</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.html#the-simpler-example" style="font-size: 80%;">The simpler Example</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs030.html#friday-september-10" style="font-size: 80%;">Friday September 10</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs031.html#linking-the-regression-analysis-with-a-statistical-interpretation" style="font-size: 80%;">Linking the regression analysis with a statistical interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#assumptions-made" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs033.html#expectation-value-and-variance" style="font-size: 80%;">Expectation value and variance</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs034.html#expectation-value-and-variance-for-boldsymbol-beta" style="font-size: 80%;">Expectation value and variance for \( \boldsymbol{\beta} \)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs035.html#deriving-ols-from-a-probability-distribution" style="font-size: 80%;">Deriving OLS from a probability distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs036.html#independent-and-identically-distrubuted-iid" style="font-size: 80%;">Independent and Identically Distrubuted (iid)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs037.html#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs038.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs039.html#more-basic-statistics-and-bayes-theorem" style="font-size: 80%;">More basic Statistics and Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs040.html#marginal-probability" style="font-size: 80%;">Marginal Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs041.html#conditional-probability" style="font-size: 80%;">Conditional Probability</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs042.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs043.html#interpretations-of-bayes-theorem" style="font-size: 80%;">Interpretations of Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs044.html#example-of-usage-of-bayes-theorem" style="font-size: 80%;">Example of Usage of Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs045.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs046.html#bayes-theorem-and-ridge-and-lasso-regression" style="font-size: 80%;">Bayes' Theorem and Ridge and Lasso Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs047.html#test-function-for-what-happens-with-ols-ridge-and-lasso" style="font-size: 80%;">Test Function for what happens with OLS, Ridge and Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs048.html#invoking-bayes-theorem" style="font-size: 80%;">Invoking Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs050.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="ridge-and-bayes" class="anchor">Ridge and Bayes </h2>
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<p>
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With the posterior probability defined by a likelihood which we have
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already modeled and an unknown prior, we are now ready to make
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additional models for the prior.
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<p>
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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
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$$
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p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
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$$
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<p>
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Our posterior probability becomes then (omitting the normalization factor which is just a constant)
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$$
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p(\boldsymbol{\beta\vert\boldsymbol{D})}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
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$$
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<p>
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We can now optimize this quantity with respect to \( \boldsymbol{\beta} \). As we
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did for OLS, this is most conveniently done by taking the negative
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logarithm of the posterior probability. Doing so and leaving out the
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constants terms that do not depend on \( \beta \), we have
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$$
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C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
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$$
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and replacing \( 1/2\tau^2 \) with \( \lambda \) we have
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$$
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C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
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$$
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which is our Ridge cost function! Nice, isn't it?
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<p>
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<p>
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</center>
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</body>
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</html>
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