mhjensen
230 lines
10 KiB
HTML
230 lines
10 KiB
HTML
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<title>Data Analysis and Machine Learning: Elements of Bayesian theory and Bayesian Neural Networks</title>
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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="#___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-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="part0011"></a>
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<!-- !split -->
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<h2 id="___sec5" class="anchor">Bayes' theorem </h2>
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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Bayes' theorem follows directly from the product rule
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$$
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$$
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p(A|B,I) = \frac{p(B|A,I) p(A|I)}{p(B|I)}.
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$$
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$$
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The importance of this property to data analysis becomes apparent if we replace \( A \) and \( B \) by hypothesis(\( H \)) and data(\( D \)):
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$$
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\begin{align}
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p(H|D,I) &= \frac{p(D|H,I) p(H|I)}{p(D|I)}.
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\tag{1}
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\end{align}
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$$
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The power of Bayes’ theorem lies in the fact that it relates the quantity of interest, the probability that the hypothesis is true given the data, to the term we have a better chance of being able to assign, the probability that we would have observed the measured data if the hypothesis was true.
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