mhjensen
229 lines
11 KiB
HTML
229 lines
11 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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<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="._Bayesian-bs001.html#___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-bs003.html#___sec2" style="font-size: 80%;"><b>Statistical 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="#___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-bs017.html#___sec9" style="font-size: 80%;"><b>Example: Is this a fair coin?</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="part0013"></a>
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<!-- !split -->
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<h2 id="___sec6" class="anchor">The friends of Bayes' theorem </h2>
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<div class="panel panel-default">
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<dl>
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<dt>Normalization:<dd>
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\( \sum_i p(H_i|\ldots) = 1 \).
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<dt>Marginalization:<dd>
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\( \sum_i p(A,H_i|I) = \sum_i p(H_i|A,I) p(A|I) = p(A|I) \).
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<dt>Marginalization (continuum limit):<dd>
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\( \int dx p(A,H(x)|I) = p(A|I) \).
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</dl>
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In the above, \( H_i \) is an exclusive and exhaustive list of hypotheses. For example,let’s imagine that there are five candidates in a presidential election; then \( H_1 \) could be the proposition that the first candidate will win, and so on. The probability that \( A \) is true, for example that unemployment will be lower in a year’s time (given all relevant information \( I \), but irrespective of whoever becomes president) is then given by \( \sum_i p(A,H_i|I) \).
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<p>
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In the continuum limit of propositions we must understand \( p(\ldots) \) as a pdf (probability density function).
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<p>
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Marginalization is a very powerful device in data analysis because it enables us to deal with nuisance parameters; that is, quantities which necessarily enter the analysis but are of no intrinsic interest. The unwanted background signal present in many experimental measurements are examples of nuisance parameters.
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<li><a href="._Bayesian-bs000.html">1</a></li>
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<li><a href="">...</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="._Bayesian-bs021.html">22</a></li>
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<li><a href="._Bayesian-bs022.html">23</a></li>
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