Bayesian inference 3 - using x and θ

xholonWorkbook.xml

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<!--Xholon Workbook http://www.primordion.com/Xholon/gwt/ MIT License, Copyright (C) Ken Webb, Fri May 09 2014 12:10:02 GMT-0400 (EDT)-->
<XholonWorkbook>

<Notes><![CDATA[
Xholon
------
Title: Bayesian inference 3 - using x and θ
Description: 
Url: http://www.primordion.com/Xholon/gwt/
InternalName: 575357d9a4c8dc0bf4e3
Keywords: 

My Notes
--------
This is associated with XholonWorkbook ceb4083319ebfa4a7bd0 and c48e278ac02dce038398

Dr James V Stone of the Psychology Department at Sheffield University in the UK, has written a book called "Bayes' Rule - A Tutorial Introduction to Bayesian Analysis". The first chapter[1] includes an extensive introductory example called "Poxy Diseases", and sections on "a more succinct notation".


References
----------
(1) http://jim-stone.staff.shef.ac.uk/BookBayes2012/bookbayesch01.pdf
(2) http://en.wikipedia.org/wiki/Bayesian_inference

]]></Notes> 

<_-.XholonClass>
  <BayesianSystem/>
  
  
  <!-- types of objects (entities) in the statistician Thomas Bayes' domain [2] -->
  <BayesianEntity>
    
    <HypothesisBE> <!-- H "H stands for any hypothesis whose probability may be affected by data (called evidence below). Often there are competing hypotheses, from which one chooses the most probable." -->
      <Smallpox/> <!-- θ_s  hypothesis that the patient's disease is smallpox -->
      <Chickenpox/> <!-- θ_c  hypothesis that the patient's disease is chickenpox -->
    </HypothesisBE>
    
    <EvidenceBE> <!-- E data "the evidence E corresponds to new data that were not used in computing the prior probability" -->
      <Spots/> <!-- x  evidence that the patient's symptoms are spots -->
    </EvidenceBE>
    
    <AntecedentBE>
      <PriorProbabilityBE/> <!-- P(H) "the prior probability, is the probability of H before E is observed. This indicates one's previous estimate of the probability that a hypothesis is true, before gaining the current evidence." -->
      
      <LikelihoodBE/> <!-- P(E|H) "the probability of observing E given H. As a function of E with H fixed, this is the likelihood. The likelihood function should not be confused with P(H | E) as a function of H rather than of E. It indicates the compatibility of the evidence with the given hypothesis." -->
    </AntecedentBE>
    
    <PosteriorProbabilityBE/> <!-- P(H|E) "the posterior probability, is the probability of H given E, i.e., after E is observed. This tells us what we want to know: the probability of a hypothesis given the observed evidence." -->
    
    <MarginalLikelihoodBE/> <!-- P(E) "P(E) is sometimes termed the marginal likelihood or 'model evidence'. This factor is the same for all possible hypotheses being considered. (This can be seen by the fact that the hypothesis H does not appear anywhere in the symbol, unlike for all the other factors.) This means that this factor does not enter into determining the relative probabilities of different hypotheses." -->
    
    <PosteriorRatioBE/> <!-- ratio between two (or more?) posterior probabilities -->
    
    <!-- collections, containers -->
    <HypothesesBE/>
    <EvidencesBE/> <!-- symptoms -->
    <ModelsBE/> <!-- diseases, a set of candidate models -->
    <ModelBE/> <!-- disease -->
  
  </BayesianEntity>
  
</_-.XholonClass>

<xholonClassDetails>
  <LikelihoodBE>
    <port name="evidence" connector="../../../EvidencesBE/Spots"/>
    <port name="hypothesis" connector=".[@roleName='p(x|θ_c)']/../../../HypothesesBE/Chickenpox"/>
    <port name="hypothesis" connector=".[@roleName='p(x|θ_s)']/../../../HypothesesBE/Smallpox"/>
  </LikelihoodBE>
  
  <PriorProbabilityBE>
    <port name="hypothesis" connector=".[@roleName='p(θ_c)']/../../../HypothesesBE/Chickenpox"/>
    <port name="hypothesis" connector=".[@roleName='p(θ_s)']/../../../HypothesesBE/Smallpox"/>
  </PriorProbabilityBE>
  
  <MarginalLikelihoodBE>
    <port name="evidence" connector="../../../EvidencesBE/Spots"/>
  </MarginalLikelihoodBE>
  
  <PosteriorProbabilityBE>
    <port name="hypothesis" connector=".[@roleName='p(θ_c|x)']/../../../HypothesesBE/Chickenpox"/>
    <port name="hypothesis" connector=".[@roleName='p(θ_s|x)']/../../../HypothesesBE/Smallpox"/>
    <port name="evidence" connector="../../../EvidencesBE/Spots"/>
  </PosteriorProbabilityBE>
</xholonClassDetails>

<BayesianSystem>
  <!-- see Figure 1.4 [1] -->
  
  <HypothesesBE>
    <Chickenpox/> <!-- θ_c -->
    <Smallpox/> <!-- θ_s -->
  </HypothesesBE>
  
  <EvidencesBE>
    <Spots/> <!-- x -->
  </EvidencesBE>
  
  <ModelsBE>
    <!-- the goal is to select the most probable of these candidate models, this is called 'model selection' -->
    
    <ModelBE>
      <LikelihoodBE roleName="p(x|θ_c)" val="0.8"/>
      <PriorProbabilityBE roleName="p(θ_c)" val="0.1"/>
      <MarginalLikelihoodBE roleName="p(x)" val="0.081"/>
      <PosteriorProbabilityBE roleName="p(θ_c|x)"/> <!-- calculated result should be 0.988 -->
    </ModelBE>
    
    <ModelBE>
      <LikelihoodBE roleName="p(x|θ_s)" val="0.9"/> <!-- maximum likelihood estimate (MLE) -->
      <PriorProbabilityBE roleName="p(θ_s)" val="0.001"/>
      <MarginalLikelihoodBE roleName="p(x)" val="0.081"/>
      <PosteriorProbabilityBE roleName="p(θ_s|x)"/> <!-- calculated result should be 0.011 -->
    </ModelBE>
    
  </ModelsBE>
  
  <PosteriorRatioBE/>
  
</BayesianSystem>

<PosteriorProbabilityBEbehavior implName="org.primordion.xholon.base.Behavior_gwtjs"><![CDATA[
  var posterior;
  var beh = {
    postConfigure: function() {
      posterior = this.cnode.parent();
      var likelihood = posterior.parent().first();
      var prior = likelihood.next();
      var marginalLikelihood = prior.next();
      posterior.val((likelihood.val() * prior.val() / marginalLikelihood.val()));
      posterior.println(posterior.xhc().name() + " " + posterior.role() + " = "
        + posterior.val().toFixed(3));
    }
  }
]]></PosteriorProbabilityBEbehavior>

<PosteriorRatioBEbehavior implName="org.primordion.xholon.base.Behavior_gwtjs"><![CDATA[
  var pratio;
  var beh = {
    postConfigure: function() {
      pratio = this.cnode.parent();
      var pprob1 = pratio.xpath("../ModelsBE/ModelBE[1]/PosteriorProbabilityBE");
      var pprob2 = pratio.xpath("../ModelsBE/ModelBE[2]/PosteriorProbabilityBE");
      pratio.val(pprob1.val() / pprob2.val());
      pratio.println(pratio.xhc().name() + " " + pprob1.role() + " / "
        + pprob2.role() + " = " + pratio.val().toFixed(1));
      pratio.println("In summary:\n  It's about " + Math.ceil(pratio.val())
        + " times more probable that the patient has " + pprob1.hypothesis.xhc().name()
        + " rather than " + pprob2.hypothesis.xhc().name() + ".");
    }
  }
]]></PosteriorRatioBEbehavior>

<SvgClient><Attribute_String roleName="svgUri"><![CDATA[data:image/svg+xml,
<svg width="100" height="50" xmlns="http://www.w3.org/2000/svg">
  <g>
    <title>Diseases</title>
    <rect id="BayesianSystem/ModelsBE" fill="#98FB98" height="50" width="50" x="25" y="0"/>
    <g>
      <title>Smallpox</title>
      <rect id="BayesianSystem/HypothesesBE/Smallpox" fill="#6AB06A" height="50" width="10" x="80" y="0"/>
    </g>
  </g>
</svg>
]]></Attribute_String><Attribute_String roleName="setup">${MODELNAME_DEFAULT},${SVGURI_DEFAULT}</Attribute_String></SvgClient>

</XholonWorkbook>