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| 63 | <h1><a class="anchor" name="intro">Introduction to Bayesian Decision Making Toolbox BDM </a></h1>This is a brief introduction into elements used in the BDM. The toolbox was designed for two principle tasks:<p> |
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| 64 | <ul> |
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| 65 | <li> |
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| 66 | Design of Bayesian decisions-making startegies, </li> |
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| 67 | <li> |
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| 68 | Bayesian system identification for on-line and off-line scenarios. </li> |
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| 69 | </ul> |
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| 70 | Theoretically, the latter is a special case of the former, however we list it separately to highlight its importance in practical applications.<p> |
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| 71 | Here, we describe basic objects that are required for implementation of the Bayesian parameter estimation.<p> |
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| 72 | Key objects are: <dl> |
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| 73 | <dt>Bayesian Model: class <code>BM</code> </dt> |
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| 74 | <dd>which is an encapsulation of the likelihood function, the prior and methodology of evaluation of the Bayes rule. This methodology may be either exact or approximate. </dd> |
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| 75 | <dt>Posterior density of the parameter: class <code>epdf</code> </dt> |
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| 76 | <dd>representing posterior density of the parameter. Methods defined on this class allow any manipulation of the posterior, such as moment evaluation, marginalization and conditioning. </dd> |
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| 77 | </dl> |
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| 78 | <h2><a class="anchor" name="bm"> |
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| 79 | Class BM</a></h2> |
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| 80 | The class BM is designed for both on-line and off-line estimation. We make the following assumptions about data: <ul> |
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| 81 | <li> |
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| 82 | an individual data record is stored in a vector, <code>vec</code> <code>dt</code>, </li> |
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| 83 | <li> |
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| 84 | a set of data records is stored in a matrix,<code>mat</code> <code>D</code>, where each column represent one individual data record </li> |
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| 85 | </ul> |
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| 86 | <p> |
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| 87 | On-line estimation is implemented by method <div class="fragment"><pre class="fragment"> <span class="keywordtype">void</span> bayes(vec dt) |
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| 88 | </pre></div> Off-line estimation is implemented by method <div class="fragment"><pre class="fragment"> <span class="keywordtype">void</span> bayesB(mat D) |
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| 89 | </pre></div><p> |
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| 90 | As an intermediate product, the bayes rule computes marginal likelihood of the data records <img class="formulaInl" alt="$ f(D) $" src="form_103.png">. Numerical value of this quantity which is important e.g. for model selection can be obtained by calling method <code>_ll()</code>.<h2><a class="anchor" name="epdf"> |
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| 91 | Getting results from BM</a></h2> |
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| 92 | Class <code>BM</code> offers several ways how to obtain results: <ul> |
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| 93 | <li> |
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| 94 | generation of posterior or predictive pdfs, methods <code>_epdf()</code> and <code>predictor()</code> </li> |
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| 95 | <li> |
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| 96 | direct evaluation of predictive likelihood, method <code>logpred()</code> </li> |
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| 97 | </ul> |
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| 98 | Underscore in the name of method <code>_epdf()</code> indicate that the method returns a pointer to the internal posterior density of the model. On the other hand, <code>predictor</code> creates a new structure of type <code>epdf()</code>.<p> |
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| 99 | Direct evaluation of predictive pdfs via logpred offers a shortcut for more efficient implementation.<h2><a class="anchor" name="epdf"> |
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| 100 | Getting results from BM</a></h2> |
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| 101 | As introduced above, the results of parameter estimation are in the form of probability density function conditioned on numerical values. This type of information is represented by class <code>epdf</code>.<p> |
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| 102 | This class allows such as moment evaluation via methods <code>mean()</code> and <code>variance()</code>, marginalization via method <code>marginal()</code>, and conditioning via method <code>condition()</code>.<p> |
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| 103 | Also, it allows generation of a sample via <code>sample()</code> and evaluation of one value of the posterior parameter likelihood via <code>evallog()</code>. Multivariate versions of these operations are also available by adding suffix <code>_m</code>, i.e. <code>sample_m()</code> and <code>evallog_m()</code>. These methods providen multiple samples and evaluation of likelihood in multiple points respectively.<h2><a class="anchor" name="pc"> |
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| 104 | Classes for probability calculus</a></h2> |
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| 105 | When a more demanding task then generation of point estimate of the parameter is required, the power of general probability claculus can be used. The following classes (together with <code>epdf</code> introduced above) form the basis of the calculus: <ul> |
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| 106 | <li> |
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| 107 | <code>mpdf</code> a pdf conditioned on another symbolic variable, </li> |
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| 108 | <li> |
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| 109 | <code>RV</code> a symbolic variable on which pdfs are defined. </li> |
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| 110 | </ul> |
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| 111 | The former class is an extension of mpdf that allows conditioning on a symbolic variable. Hence, when numerical results - such as samples - are required, numericla values of the condition must be provided. The names of methods of the <code>epdf</code> are used extended by suffix <code>cond</code>, i.e. <code>samplecond()</code>, <code>evallogcond()</code>, where <code>cond</code> precedes matrix estension, i.e. <code>samplecond_m()</code> and <code>evallogcond_m()</code>.<p> |
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| 112 | The latter class is used to identify how symbolic variables are to be combined together. For example, consider the task of composition of pdfs via the chain rule: <p class="formulaDsp"> |
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| 113 | <img class="formulaDsp" alt="\[ f(a,b,c) = f(a|b,c) f(b) f(c) \]" src="form_104.png"> |
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| 114 | <p> |
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| 115 | In our setup, <img class="formulaInl" alt="$ f(a|b,c) $" src="form_105.png"> is represented by an <code>mpdf</code> while <img class="formulaInl" alt="$ f(b) $" src="form_106.png"> and <img class="formulaInl" alt="$ f(c) $" src="form_107.png"> by two <code>epdfs</code>. We need to distinguish the latter two from each other and to deside in which order they should be added to the mpdf. This distinction is facilitated by the class <code>RV</code> which uniquely identify a random varibale.<p> |
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| 116 | Therefore, each pdf keeps record on which RVs it represents; <code>epdf</code> needs to know only one <code>RV</code> stored in the attribute <code>rv</code>; <code>mpdf</code> needs to keep two <code>RVs</code>, one for variable on which it is defined (<code>rv</code>) and one for variable incondition which is stored in attribute <code>rvc</code>. </div> |
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