First draft of chapter on Gaussian processes.
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Christian Forssen
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<title>Data Analysis and Machine Learning: Machine learning with Gaussian Processes</title>
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<!-- ------------------- main content ---------------------- -->
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<center><h1>Data Analysis and Machine Learning: Machine learning with Gaussian Processes</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Christian Forssén, and Morten Hjorth-Jensen -->
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<center>
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<b>Christian Forssén</b> [1]
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</center>
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<center>
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<b>Morten Hjorth-Jensen</b> [2, 3]
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</center>
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<p>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, Chalmers University of Technology, Sweden</b></center>
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<center>[2] <b>Department of Physics, University of Oslo</b></center>
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<center>[3] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Mar 19, 2018</h4></center> <!-- date -->
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<br>
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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="___sec0">What is a Gaussian Process? </h2>
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<ul>
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<li> We have considered splines and kernel regression methods. These</li>
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</ul>
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require choice of somewhat arbitrary set of knots.
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<ul>
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<li> Antoher possibility is to setup a prior distribution for the
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regression function using a <em>Gaussian Process</em>.</li>
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<li> This is a very flexible class of models that has distinct computational
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and theoretical advantages. It can be viewed as a potentially
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infinite-dimensional generalization of Gaussian distributions.</li>
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<li> See the excellent (and free) book <a href="http://www.gaussianprocess.org/gpml/" target="_blank">Gaussian Processes for Machine
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Learning</a> by Carl Edward
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Rasmussen and Christopher K. I. Williams.</li>
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</ul>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec1">Gaussian process regression </h2>
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<ul>
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<li> Realizations from a Gaussian process correspond to random functions</li>
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<li> Let us first consider an unknown regression function \( \mu(x) \) that
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depends on a single, continuous variable \( x \).</li>
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<li> The Gaussian process is written as \( \mu \sim \mathrm{GP}(m,k) \), and
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is parametrized in terms of a mean function \( m(x) \) and a covariance
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function \( k(x,x') \).</li>
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<li> The GP prior on \( \mu \) describes it as a random function for which
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the values at any set of \( N \) prespecified points \( \{x_i\}_{i=1}^N \)
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are a draw from a \( N \)-dimensional normal distribution</li>
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</ul>
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$$
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$$
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\mu(x_1), \ldots \mu(x_N) \sim \mathrm{N}\left( \left( m(x_1),
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\ldots, m(x_N) \right), K(x_1, \ldots, x_N) \right),
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$$
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$$
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with mean \( m \) and covariance \( K \).
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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="___sec2">Topics </h2>
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<ul>
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<li> More matematical details</li>
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<li> The role of the covariance function (different kernels)</li>
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<li> multidimensional case</li>
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<li> examples.</li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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<center style="font-size:80%">
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<!-- copyright --> © 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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