mhjensen
224 lines
10 KiB
HTML
224 lines
10 KiB
HTML
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<a class="navbar-brand" href="Bayesian-bs.html">Data Analysis and Machine Learning: Elements of Bayesian theory and Bayesian Neural Networks</a>
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<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;"><b>Why Bayesian Statistics?</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs002.html#___sec1" style="font-size: 80%;"><b>Inference</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs004.html#___sec3" style="font-size: 80%;"><b>Some history</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs010.html#___sec4" style="font-size: 80%;"><b>The Bayesian recipe</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs011.html#___sec5" style="font-size: 80%;"><b>Bayes' theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs013.html#___sec6" style="font-size: 80%;"><b>The friends of Bayes' theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs014.html#___sec7" style="font-size: 80%;"><b>Inference With Parametric Models</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs016.html#___sec8" style="font-size: 80%;"><b>Illustrative examples with python code</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs018.html#___sec10" style="font-size: 80%;"><b>A few words on different priors</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs019.html#___sec11" style="font-size: 80%;"><b>Bayesian parameter estimation (single parameter)</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs020.html#___sec12" style="font-size: 80%;"> Example: Measured flux from a star</a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs020.html#___sec13" style="font-size: 80%;"> Simple Photon Counts: Frequentist Approach</a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs020.html#___sec14" style="font-size: 80%;"> Simple Photon Counts: Bayesian Approach</a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs020.html#___sec15" style="font-size: 80%;"> A note about priors</a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs020.html#___sec16" style="font-size: 80%;"> Simple Photon Counts: Bayesian approach in practice</a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs020.html#___sec17" style="font-size: 80%;"> Best estimates and confidence intervals</a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs020.html#___sec18" style="font-size: 80%;"> Simple Photon Counts: Best estimates and confidence intervals</a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs021.html#___sec19" style="font-size: 80%;"><b>Bayesian parameter estimation (multiple parameters, covariance)</b></a></li>
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<!-- navigation toc: --> <li><a href="._Bayesian-bs022.html#___sec20" style="font-size: 80%;"><b>Bayesian model selection</b></a></li>
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<a name="part0001"></a>
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<!-- !split -->
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<h2 id="___sec0" class="anchor">Why Bayesian Statistics? </h2>
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>
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We have already made ourselves familiar with elements of a statistical
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data analysis via quantities like the bias-variance tradeoff as well
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as some central distribution functions such as the Normal
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distribution, the binomial distribution and other probability
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distribution functions.
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<p>
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In essentially all the Machine Learning
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algorithms we have studied, our focus has been on a so-called
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<b>frequentist approach</b>, where knowledge of an underlying likelihood
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function has not been emphasized. Our data, whether we had a classification or a regression problem, have been our central points of departure.
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<p>
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Here we wish to merge this approach with the derivation of a likelihood function which can be used to make prediction on how our system under study evolves.
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We will venture into the realm of what is called Bayesian Neural Networks. To get an overarching view on what this entails, the following figure conveys the essential differences between a standard Neural network that we have met earlier and a Bayesian Neural Network. In order to get there, we need to present some of the basic elements of Bayesian statistics, starting with the product rule and Bayes' theorem.
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<p>
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<p>
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<li><a href="._Bayesian-bs000.html">«</a></li>
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<li><a href="._Bayesian-bs000.html">1</a></li>
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<li class="active"><a href="._Bayesian-bs001.html">2</a></li>
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<li><a href="._Bayesian-bs002.html">3</a></li>
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<li><a href="._Bayesian-bs003.html">4</a></li>
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<li><a href="._Bayesian-bs004.html">5</a></li>
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<li><a href="._Bayesian-bs005.html">6</a></li>
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<li><a href="._Bayesian-bs006.html">7</a></li>
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<li><a href="._Bayesian-bs007.html">8</a></li>
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<li><a href="._Bayesian-bs008.html">9</a></li>
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<li><a href="._Bayesian-bs009.html">10</a></li>
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<li><a href="._Bayesian-bs010.html">11</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Bayesian-bs022.html">23</a></li>
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