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    53<title>mixpp: Introduction to Bayesian Decision Making Toolbox BDM</title> 
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    67 <h1><a class="anchor" id="intro">Introduction to Bayesian Decision Making Toolbox BDM </a></h1><p>This is a brief introduction into elements used in the BDM. The toolbox was designed for two principle tasks:</p> 
     62<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> 
    6863<ul> 
    6964<li> 
    … …  
    7267Bayesian system identification for on-line and off-line scenarios.  </li> 
    7368</ul> 
    74 <p>Theoretically, the latter is a special case of the former, however we list it separately to highlight its importance in practical applications.</p> 
    75 <p>Here, we describe basic objects that are required for implementation of the Bayesian parameter estimation.</p> 
    76 <p>Key objects are: </p> 
    77 <dl> 
     69Theoretically, the latter is a special case of the former, however we list it separately to highlight its importance in practical applications.<p> 
     70Here, we describe basic objects that are required for implementation of the Bayesian parameter estimation.<p> 
     71Key objects are: <dl> 
    7872<dt>Bayesian Model: class <code>BM</code>  </dt> 
    7973<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> 
    … …  
    8175<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> 
    8276</dl> 
    83 <h2><a class="anchor" id="bm"> 
     77<h2><a class="anchor" name="bm"> 
    8478Class BM</a></h2> 
    85 <p>The class BM is designed for both on-line and off-line estimation. We make the following assumptions about data: </p> 
    86 <ul> 
     79The class BM is designed for both on-line and off-line estimation. We make the following assumptions about data: <ul> 
    8780<li> 
    8881an individual data record is stored in a vector, <code>vec</code> <code>dt</code>,  </li> 
    … …  
    9083a set of data records is stored in a matrix,<code>mat</code> <code>D</code>, where each column represent one individual data record  </li> 
    9184</ul> 
    92 <p>On-line estimation is implemented by method </p> 
    93 <div class="fragment"><pre class="fragment"> <span class="keywordtype">void</span> bayes(vec dt) 
    94 </pre></div><p> Off-line estimation is implemented by method </p> 
    95 <div class="fragment"><pre class="fragment"> <span class="keywordtype">void</span> bayesB(mat D) 
    96 </pre></div><p>As an intermediate product, the bayes rule computes marginal likelihood of the data records <img class="formulaInl" alt="$ f(D) $" src="form_114.png"/>. Numerical value of this quantity which is important e.g. for model selection can be obtained by calling method <code>_ll()</code>.</p> 
    97 <h2><a class="anchor" id="epdf"> 
     85<p> 
     86On-line estimation is implemented by method <div class="fragment"><pre class="fragment"> <span class="keywordtype">void</span> bayes(vec dt) 
     87</pre></div> Off-line estimation is implemented by method <div class="fragment"><pre class="fragment"> <span class="keywordtype">void</span> bayesB(mat D) 
     88</pre></div><p> 
     89As an intermediate product, the bayes rule computes marginal likelihood of the data records <img class="formulaInl" alt="$ f(D) $" src="form_114.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"> 
    9890Getting results from BM</a></h2> 
    99 <p>Class <code>BM</code> offers several ways how to obtain results: </p> 
    100 <ul> 
     91Class <code>BM</code> offers several ways how to obtain results: <ul> 
    10192<li> 
    10293generation of posterior or predictive pdfs, methods <code>_epdf()</code> and <code>predictor()</code>  </li> 
    … …  
    10495direct evaluation of predictive likelihood, method <code>logpred()</code>  </li> 
    10596</ul> 
    106 <p>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> 
    107 <p>Direct evaluation of predictive pdfs via logpred offers a shortcut for more efficient implementation.</p> 
    108 <h2><a class="anchor" id="epdf"> 
     97Underscore 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> 
     98Direct evaluation of predictive pdfs via logpred offers a shortcut for more efficient implementation.<h2><a class="anchor" name="epdf"> 
    10999Getting results from BM</a></h2> 
    110 <p>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> 
    111 <p>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> 
    112 <p>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.</p> 
    113 <h2><a class="anchor" id="pc"> 
     100As 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> 
     101This 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> 
     102Also, 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"> 
    114103Classes for probability calculus</a></h2> 
    115 <p>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: </p> 
    116 <ul> 
     104When 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> 
    117105<li> 
    118106<code>mpdf</code> a pdf conditioned on another symbolic variable, </li> 
    … …  
    120108<code>RV</code> a symbolic variable on which pdfs are defined. </li> 
    121109</ul> 
    122 <p>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> 
    123 <p>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> 
    124 <p class="formulaDsp"> 
    125 <img class="formulaDsp" alt="\[ f(a,b,c) = f(a|b,c) f(b) f(c) \]" src="form_115.png"/> 
    126 </p> 
    127 <p> In our setup, <img class="formulaInl" alt="$ f(a|b,c) $" src="form_116.png"/> is represented by an <code>mpdf</code> while <img class="formulaInl" alt="$ f(b) $" src="form_117.png"/> and <img class="formulaInl" alt="$ f(c) $" src="form_118.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> 
    128 <p>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>. </p> 
    129 </div> 
    130 <hr size="1"/><address style="text-align: right;"><small>Generated on Sun Sep 27 00:49:04 2009 for mixpp by&nbsp; 
     110The 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> 
     111The 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"> 
     112<img class="formulaDsp" alt="\[ f(a,b,c) = f(a|b,c) f(b) f(c) \]" src="form_115.png"> 
     113<p> 
     114 In our setup, <img class="formulaInl" alt="$ f(a|b,c) $" src="form_116.png"> is represented by an <code>mpdf</code> while <img class="formulaInl" alt="$ f(b) $" src="form_117.png"> and <img class="formulaInl" alt="$ f(c) $" src="form_118.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> 
     115Therefore, 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> 
     116<hr size="1"><address style="text-align: right;"><small>Generated on Wed Oct 7 17:34:44 2009 for mixpp by&nbsp; 
    131117<a href="http://www.doxygen.org/index.html"> 
    132 <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.6.1 </small></address> 
     118<img src="doxygen.png" alt="doxygen" align="middle" border="0"></a> 1.5.9 </small></address> 
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