added Bayes stuff
This commit is contained in:
Morten Hjorth-Jensen
@@ -155,6 +155,10 @@ Automatically generated HTML file from DocOnce source
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None,
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'example-of-usage-of-bayes-theorem'),
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('Doing it correctly', 2, None, 'doing-it-correctly'),
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("Bayes' Theorem and Ridge and Lasso Regression",
|
||||
2,
|
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None,
|
||||
'bayes-theorem-and-ridge-and-lasso-regression'),
|
||||
('Why resampling methods', 2, None, 'why-resampling-methods'),
|
||||
('Resampling methods', 2, None, 'resampling-methods'),
|
||||
('Resampling approaches can be computationally expensive',
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||||
@@ -323,33 +327,34 @@ MathJax.Hub.Config({
|
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<!-- navigation toc: --> <li><a href="._week36-bs041.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-bs042.html#example-of-usage-of-bayes-theorem" style="font-size: 80%;">Example of Usage of Bayes' theorem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs043.html#doing-it-correctly" style="font-size: 80%;">Doing it correctly</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs047.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs049.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs046.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs047.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs048.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs049.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs050.html#resampling-methods-jackknife-and-bootstrap" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs051.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs052.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs053.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs054.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs055.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs056.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs057.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs058.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs059.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs060.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>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs061.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs062.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>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs063.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs064.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs065.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs066.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs067.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs068.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>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs069.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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||||
<!-- navigation toc: --> <li><a href="._week36-bs070.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs044.html#bayes-theorem-and-ridge-and-lasso-regression" style="font-size: 80%;">Bayes' Theorem and Ridge and Lasso Regression</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs048.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs050.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs047.html#resampling-approaches-can-be-computationally-expensive" style="font-size: 80%;">Resampling approaches can be computationally expensive</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs048.html#why-resampling-methods" style="font-size: 80%;">Why resampling methods ?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs049.html#statistical-analysis" style="font-size: 80%;">Statistical analysis</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs050.html#resampling-methods" style="font-size: 80%;">Resampling methods</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs051.html#resampling-methods-jackknife-and-bootstrap" style="font-size: 80%;">Resampling methods: Jackknife and Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs052.html#resampling-methods-jackknife" style="font-size: 80%;">Resampling methods: Jackknife</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs053.html#jackknife-code-example" style="font-size: 80%;">Jackknife code example</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs054.html#resampling-methods-bootstrap" style="font-size: 80%;">Resampling methods: Bootstrap</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs055.html#resampling-methods-bootstrap-background" style="font-size: 80%;">Resampling methods: Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs056.html#resampling-methods-more-bootstrap-background" style="font-size: 80%;">Resampling methods: More Bootstrap background</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs057.html#resampling-methods-bootstrap-approach" style="font-size: 80%;">Resampling methods: Bootstrap approach</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs058.html#resampling-methods-bootstrap-steps" style="font-size: 80%;">Resampling methods: Bootstrap steps</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs059.html#code-example-for-the-bootstrap-method" style="font-size: 80%;">Code example for the Bootstrap method</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs060.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs061.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>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs062.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs063.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>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs064.html#the-bias-variance-tradeoff" style="font-size: 80%;">The bias-variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs065.html#example-code-for-bias-variance-tradeoff" style="font-size: 80%;">Example code for Bias-Variance tradeoff</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs066.html#understanding-what-happens" style="font-size: 80%;">Understanding what happens</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs067.html#summing-up" style="font-size: 80%;">Summing up</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs068.html#another-example-from-scikit-learn-s-repository" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs069.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>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs070.html#the-same-example-but-now-with-cross-validation" style="font-size: 80%;">The same example but now with cross-validation</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week36-bs071.html#cross-validation-with-ridge" style="font-size: 80%;">Cross-validation with Ridge</a></li>
|
||||
|
||||
</ul>
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||||
</li>
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@@ -408,7 +413,7 @@ MathJax.Hub.Config({
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<li><a href="._week36-bs008.html">9</a></li>
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<li><a href="._week36-bs009.html">10</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week36-bs070.html">71</a></li>
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<li><a href="._week36-bs071.html">72</a></li>
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<li><a href="._week36-bs001.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -1637,9 +1637,9 @@ Let us try to illustrate Bayes' theorem through an example.
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<h2 id="example-of-usage-of-bayes-theorem">Example of Usage of Bayes' theorem </h2>
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<p>
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Let us suppose that you are undergoing a series of mammography scan in
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Let us suppose that you are undergoing a series of mammography scans in
|
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order to rule out possible breast cancer cases. We define the
|
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sensitivity for a positive event by the variable \( X \) (it takes binary
|
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sensitivity for a positive event by the variable \( X \). It takes binary
|
||||
values with \( X=1 \) representing a positive event and \( X=0 \) being a
|
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negative event. We reserve \( Y \) as a classification parameter for
|
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either a negative or a postive breast cancer confirmation.
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@@ -1657,7 +1657,7 @@ $$
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<p> <br>
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<p>
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This obviously sounds scaring since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer.
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This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer.
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It is however not correct, as the following Bayesian analysis shows.
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</section>
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@@ -1696,6 +1696,20 @@ That is, in case of a positive test, there is only a \( 3\% \) chance of having
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</section>
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<section>
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<h2 id="bayes-theorem-and-ridge-and-lasso-regression">Bayes' Theorem and Ridge and Lasso Regression </h2>
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<p>
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Hitherto we have discussed Ridge and Lasso regression in terms of a
|
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linear analysis. This may to many of you feel rather technical and
|
||||
perhaps not that intuitive. The question is whether we can develop a
|
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more intuitive way of understanding what Ridge and Lasso express.
|
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<p>
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Before we proceed let us perform a Ridge and OLS analysis of a polynomial fit.
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</section>
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<section>
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<h2 id="why-resampling-methods">Why resampling methods </h2>
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@@ -175,6 +175,10 @@ div { text-align: justify; text-justify: inter-word; }
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None,
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'example-of-usage-of-bayes-theorem'),
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('Doing it correctly', 2, None, 'doing-it-correctly'),
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("Bayes' Theorem and Ridge and Lasso Regression",
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2,
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None,
|
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'bayes-theorem-and-ridge-and-lasso-regression'),
|
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('Why resampling methods', 2, None, 'why-resampling-methods'),
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('Resampling methods', 2, None, 'resampling-methods'),
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('Resampling approaches can be computationally expensive',
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@@ -1629,9 +1633,9 @@ Let us try to illustrate Bayes' theorem through an example.
|
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<h2 id="example-of-usage-of-bayes-theorem">Example of Usage of Bayes' theorem </h2>
|
||||
|
||||
<p>
|
||||
Let us suppose that you are undergoing a series of mammography scan in
|
||||
Let us suppose that you are undergoing a series of mammography scans in
|
||||
order to rule out possible breast cancer cases. We define the
|
||||
sensitivity for a positive event by the variable \( X \) (it takes binary
|
||||
sensitivity for a positive event by the variable \( X \). It takes binary
|
||||
values with \( X=1 \) representing a positive event and \( X=0 \) being a
|
||||
negative event. We reserve \( Y \) as a classification parameter for
|
||||
either a negative or a postive breast cancer confirmation.
|
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@@ -1647,7 +1651,7 @@ p(X=1\vert Y=1) =0.8.
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$$
|
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|
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<p>
|
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This obviously sounds scaring since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer.
|
||||
This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer.
|
||||
It is however not correct, as the following Bayesian analysis shows.
|
||||
|
||||
<p>
|
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@@ -1681,6 +1685,20 @@ That is, in case of a positive test, there is only a \( 3\% \) chance of having
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="bayes-theorem-and-ridge-and-lasso-regression">Bayes' Theorem and Ridge and Lasso Regression </h2>
|
||||
|
||||
<p>
|
||||
Hitherto we have discussed Ridge and Lasso regression in terms of a
|
||||
linear analysis. This may to many of you feel rather technical and
|
||||
perhaps not that intuitive. The question is whether we can develop a
|
||||
more intuitive way of understanding what Ridge and Lasso express.
|
||||
|
||||
<p>
|
||||
Before we proceed let us perform a Ridge and OLS analysis of a polynomial fit.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
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|
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<h2 id="why-resampling-methods">Why resampling methods </h2>
|
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|
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<p>
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@@ -180,6 +180,10 @@ div { text-align: justify; text-justify: inter-word; }
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None,
|
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'example-of-usage-of-bayes-theorem'),
|
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('Doing it correctly', 2, None, 'doing-it-correctly'),
|
||||
("Bayes' Theorem and Ridge and Lasso Regression",
|
||||
2,
|
||||
None,
|
||||
'bayes-theorem-and-ridge-and-lasso-regression'),
|
||||
('Why resampling methods', 2, None, 'why-resampling-methods'),
|
||||
('Resampling methods', 2, None, 'resampling-methods'),
|
||||
('Resampling approaches can be computationally expensive',
|
||||
@@ -1634,9 +1638,9 @@ Let us try to illustrate Bayes' theorem through an example.
|
||||
<h2 id="example-of-usage-of-bayes-theorem">Example of Usage of Bayes' theorem </h2>
|
||||
|
||||
<p>
|
||||
Let us suppose that you are undergoing a series of mammography scan in
|
||||
Let us suppose that you are undergoing a series of mammography scans in
|
||||
order to rule out possible breast cancer cases. We define the
|
||||
sensitivity for a positive event by the variable \( X \) (it takes binary
|
||||
sensitivity for a positive event by the variable \( X \). It takes binary
|
||||
values with \( X=1 \) representing a positive event and \( X=0 \) being a
|
||||
negative event. We reserve \( Y \) as a classification parameter for
|
||||
either a negative or a postive breast cancer confirmation.
|
||||
@@ -1652,7 +1656,7 @@ p(X=1\vert Y=1) =0.8.
|
||||
$$
|
||||
|
||||
<p>
|
||||
This obviously sounds scaring since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer.
|
||||
This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of \( 80\% \) for having cancer.
|
||||
It is however not correct, as the following Bayesian analysis shows.
|
||||
|
||||
<p>
|
||||
@@ -1686,6 +1690,20 @@ That is, in case of a positive test, there is only a \( 3\% \) chance of having
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="bayes-theorem-and-ridge-and-lasso-regression">Bayes' Theorem and Ridge and Lasso Regression </h2>
|
||||
|
||||
<p>
|
||||
Hitherto we have discussed Ridge and Lasso regression in terms of a
|
||||
linear analysis. This may to many of you feel rather technical and
|
||||
perhaps not that intuitive. The question is whether we can develop a
|
||||
more intuitive way of understanding what Ridge and Lasso express.
|
||||
|
||||
<p>
|
||||
Before we proceed let us perform a Ridge and OLS analysis of a polynomial fit.
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="why-resampling-methods">Why resampling methods </h2>
|
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|
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<p>
|
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|
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Binary file not shown.
@@ -2052,9 +2052,9 @@
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"\n",
|
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"## Example of Usage of Bayes' theorem\n",
|
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"\n",
|
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"Let us suppose that you are undergoing a series of mammography scan in\n",
|
||||
"Let us suppose that you are undergoing a series of mammography scans in\n",
|
||||
"order to rule out possible breast cancer cases. We define the\n",
|
||||
"sensitivity for a positive event by the variable $X$ (it takes binary\n",
|
||||
"sensitivity for a positive event by the variable $X$. It takes binary\n",
|
||||
"values with $X=1$ representing a positive event and $X=0$ being a\n",
|
||||
"negative event. We reserve $Y$ as a classification parameter for\n",
|
||||
"either a negative or a postive breast cancer confirmation.\n",
|
||||
@@ -2077,7 +2077,7 @@
|
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"cell_type": "markdown",
|
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"metadata": {},
|
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"source": [
|
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"This obviously sounds scaring since many would conclude that if the test is positive, there is a likelihood of $80\\%$ for having cancer.\n",
|
||||
"This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of $80\\%$ for having cancer.\n",
|
||||
"It is however not correct, as the following Bayesian analysis shows.\n",
|
||||
"\n",
|
||||
"## Doing it correctly\n",
|
||||
@@ -2133,6 +2133,18 @@
|
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"source": [
|
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"That is, in case of a positive test, there is only a $3\\%$ chance of having cancer!\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Bayes' Theorem and Ridge and Lasso Regression\n",
|
||||
"\n",
|
||||
"Hitherto we have discussed Ridge and Lasso regression in terms of a\n",
|
||||
"linear analysis. This may to many of you feel rather technical and\n",
|
||||
"perhaps not that intuitive. The question is whether we can develop a\n",
|
||||
"more intuitive way of understanding what Ridge and Lasso express.\n",
|
||||
"\n",
|
||||
"Before we proceed let us perform a Ridge and OLS analysis of a polynomial fit. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Why resampling methods\n",
|
||||
"\n",
|
||||
"Before we proceed, we need to rethink what we have been doing. In our\n",
|
||||
|
||||
@@ -1270,9 +1270,9 @@ Let us try to illustrate Bayes' theorem through an example.
|
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!split
|
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===== Example of Usage of Bayes' theorem =====
|
||||
|
||||
Let us suppose that you are undergoing a series of mammography scan in
|
||||
Let us suppose that you are undergoing a series of mammography scans in
|
||||
order to rule out possible breast cancer cases. We define the
|
||||
sensitivity for a positive event by the variable $X$ (it takes binary
|
||||
sensitivity for a positive event by the variable $X$. It takes binary
|
||||
values with $X=1$ representing a positive event and $X=0$ being a
|
||||
negative event. We reserve $Y$ as a classification parameter for
|
||||
either a negative or a postive breast cancer confirmation.
|
||||
@@ -1287,7 +1287,7 @@ p(X=1\vert Y=1) =0.8.
|
||||
\]
|
||||
!et
|
||||
|
||||
This obviously sounds scaring since many would conclude that if the test is positive, there is a likelihood of $80\%$ for having cancer.
|
||||
This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of $80\%$ for having cancer.
|
||||
It is however not correct, as the following Bayesian analysis shows.
|
||||
|
||||
!split
|
||||
@@ -1318,6 +1318,19 @@ p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=
|
||||
!et
|
||||
That is, in case of a positive test, there is only a $3\%$ chance of having cancer!
|
||||
|
||||
|
||||
!split
|
||||
===== Bayes' Theorem and Ridge and Lasso Regression =====
|
||||
|
||||
Hitherto we have discussed Ridge and Lasso regression in terms of a
|
||||
linear analysis. This may to many of you feel rather technical and
|
||||
perhaps not that intuitive. The question is whether we can develop a
|
||||
more intuitive way of understanding what Ridge and Lasso express.
|
||||
|
||||
Before we proceed let us perform a Ridge and OLS analysis of a polynomial fit.
|
||||
|
||||
|
||||
|
||||
!split
|
||||
===== Why resampling methods =====
|
||||
|
||||
|
||||
Reference in New Issue
Block a user