Morten Hjorth-Jensen
355 lines
19 KiB
HTML
355 lines
19 KiB
HTML
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('Thursday September 16, Summary of Ridge and Lasso Regression '
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<a class="navbar-brand" href="week37-bs.html">Week 37: Summary of Ridge and Lasso Regression and Resampling Methods</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="._week37-bs001.html#plans-for-week-37" style="font-size: 80%;">Plans for week 37</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs002.html#thursday-september-16-summary-of-ridge-and-lasso-regression-and-start-resampling-methods" style="font-size: 80%;">Thursday September 16, Summary of Ridge and Lasso Regression and start Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs003.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="._week37-bs004.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="#maximum-likelihood-estimation-mle" style="font-size: 80%;">Maximum Likelihood Estimation (MLE)</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs006.html#a-new-cost-function" style="font-size: 80%;">A new Cost Function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs007.html#bayes-theorem" style="font-size: 80%;">Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.html#interpretations-of-bayes-theorem" style="font-size: 80%;">Interpretations of Bayes' Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.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="._week37-bs010.html#rerunning-the-above-code" style="font-size: 80%;">Rerunning the above code</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs011.html#invoking-bayes-theorem" style="font-size: 80%;">Invoking Bayes' theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs012.html#ridge-and-bayes" style="font-size: 80%;">Ridge and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#lasso-and-bayes" style="font-size: 80%;">Lasso and Bayes</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs016.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#resampling-methods-jackknife-and-bootstrap" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs022.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs023.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs024.html#the-central-limit-theorem" style="font-size: 80%;">The Central Limit Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs025.html#finding-the-limit" style="font-size: 80%;">Finding the Limit</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.html#rewriting-the-delta-function" style="font-size: 80%;">Rewriting the \( \delta \)-function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs027.html#identifying-terms" style="font-size: 80%;">Identifying Terms</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs028.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs029.html#confidence-intervals" style="font-size: 80%;">Confidence Intervals</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs030.html#standard-approach-based-on-the-normal-distribution" style="font-size: 80%;">Standard Approach based on the Normal Distribution</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs031.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs032.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs033.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs034.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs035.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs036.html#plotting-the-histogram" style="font-size: 80%;">Plotting the Histogram</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs037.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs038.html#a-way-to-read-the-bias-variance-tradeoff" style="font-size: 80%;">A way to Read the Bias-Variance Tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs039.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs040.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs041.html#summing-up" style="font-size: 80%;">Summing up</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs042.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs043.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs044.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs045.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs046.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs047.html#more-examples-on-bootstrap-and-cross-validation-and-errors" style="font-size: 80%;">More examples on bootstrap and cross-validation and errors</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs048.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</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>
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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="part0005"></a>
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<!-- !split -->
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<h2 id="maximum-likelihood-estimation-mle" class="anchor">Maximum Likelihood Estimation (MLE) </h2>
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<p>
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In statistics, maximum likelihood estimation (MLE) is a method of
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estimating the parameters of an assumed probability distribution,
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given some observed data. This is achieved by maximizing a likelihood
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function so that, under the assumed statistical model, the observed
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data is the most probable.
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<p>
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We will assume here that our events are given by the above Gaussian
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distribution and we will determine the optimal parameters \( \beta \) by
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maximizing the above PDF. However, computing the derivatives of a
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product function is cumbersome and can easily lead to overflow and/or
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underflowproblems, with potentials for loss of numerical precision.
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<p>
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In practice, it is more convenient to maximize the logarithm of the
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PDF because it is a monotonically increasing function of the argument.
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Alternatively, and this will be our option, we will minimize the
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negative of the logarithm since this is a monotonically decreasing
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function.
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<p>
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Note also that maximization/minimization of the logarithm of the PDF
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is equivalent to the maximization/minimization of the function itself.
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<p>
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<p>
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<li><a href="._week37-bs004.html">«</a></li>
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<li><a href="._week37-bs010.html">11</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week37-bs048.html">49</a></li>
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