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Bayesian inference 4
<?xml version="1.0" encoding="UTF-8"?>
<!--Xholon Workbook http://www.primordion.com/Xholon/gwt/ MIT License, Copyright (C) Ken Webb, Wed May 14 2014 07:30:28 GMT-0400 (EDT)-->
<XholonWorkbook>
<Notes><![CDATA[
Xholon
------
Title: Bayesian inference 4
Description:
Url: http://www.primordion.com/Xholon/gwt/
InternalName: 8688253668d5fa489b4f
Keywords:
My Notes
--------
This is associated with XholonWorkbooks ceb4083319ebfa4a7bd0 and c48e278ac02dce038398,
and is based on "Example 2: Forkandles" from "Bayes' Rule - A Tutorial Introduction to Bayesian Analysis" by Dr James V Stone [1].
If you walked into a hardware store and asked, "Have you got fork
handles?", then you would be surprised to be presented with four
candles. Even though the phrases "fork handles" and "four candles" are
acoustically almost identical, the shop assistant knows that he sells
many more candles than fork handles (Figure 1.5). This in turn, means
that he probably does not even hear the words "fork handles", but instead
hears "four candles". What has this got to do with Bayes' rule?
References
----------
(1) http://jim-stone.staff.shef.ac.uk/BookBayes2012/bookbayesch01.pdf
(2) http://en.wikipedia.org/wiki/Bayesian_inference
(3) http://en.wikipedia.org/wiki/Bayes%27_rule
]]></Notes>
<_-.XholonClass>
<BayesianSystem/>
<!-- types of objects (entities) -->
<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." -->
<FourCandles/> <!-- hypothesis that the person was saying "four candles" -->
<ForkHandles/> <!-- hypothesis that the person was saying "fork handles" -->
</HypothesisBE>
<EvidenceBE> <!-- E data "the evidence E corresponds to new data that were not used in computing the prior probability" -->
<AcousticData/> <!-- evidence from the person's acoustic data -->
</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/>
<ModelsBE/> <!-- a set of candidate models -->
<ModelBE/>
</BayesianEntity>
</_-.XholonClass>
<xholonClassDetails>
<LikelihoodBE>
<port name="evidence" connector="../../../EvidencesBE/AcousticData"/>
<port name="hypothesis" connector=".[@roleName='p(acoustic data|four candles)']/../../../HypothesesBE/FourCandles"/>
<port name="hypothesis" connector=".[@roleName='p(acoustic data|fork handles)']/../../../HypothesesBE/ForkHandles"/>
</LikelihoodBE>
<PriorProbabilityBE>
<port name="hypothesis" connector=".[@roleName='p(four candles)']/../../../HypothesesBE/FourCandles"/>
<port name="hypothesis" connector=".[@roleName='p(fork handles)']/../../../HypothesesBE/ForkHandles"/>
</PriorProbabilityBE>
<MarginalLikelihoodBE>
<port name="evidence" connector="../../../EvidencesBE/AcousticData"/>
</MarginalLikelihoodBE>
<PosteriorProbabilityBE>
<port name="hypothesis" connector=".[@roleName='p(four candles|acoustic data)']/../../../HypothesesBE/FourCandles"/>
<port name="hypothesis" connector=".[@roleName='p(fork handles|acoustic data)']/../../../HypothesesBE/ForkHandles"/>
<port name="evidence" connector="../../../EvidencesBE/AcousticData"/>
</PosteriorProbabilityBE>
</xholonClassDetails>
<BayesianSystem>
<!-- see Figure 1.7 [1] -->
<HypothesesBE>
<FourCandles/>
<ForkHandles/>
</HypothesesBE>
<EvidencesBE>
<AcousticData/>
</EvidencesBE>
<ModelsBE>
<!-- the goal is to select the most probable of these candidate models, this is called 'model selection' -->
<ModelBE>
<LikelihoodBE roleName="p(acoustic data|four candles)" val="0.6"/>
<PriorProbabilityBE roleName="p(four candles)" val="0.9"/>
<MarginalLikelihoodBE roleName="p(acoustic data)" val="0.061"/>
<PosteriorProbabilityBE roleName="p(four candles|acoustic data)"/> <!-- calculated result should be 0.885 -->
</ModelBE>
<ModelBE>
<LikelihoodBE roleName="p(acoustic data|fork handles)" val="0.7"/> <!-- maximum likelihood estimate (MLE) -->
<PriorProbabilityBE roleName="p(fork handles)" val="0.1"/>
<MarginalLikelihoodBE roleName="p(acoustic data)" val="0.061"/>
<PosteriorProbabilityBE roleName="p(fork handles|acoustic data)"/> <!-- calculated result should be 0.115 -->
</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 person said " + pprob1.hypothesis.xhc().name()
+ " rather than " + pprob2.hypothesis.xhc().name() + ".");
}
}
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>Bayesian system</text
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>Posterior ratio</text
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>Models</text
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>Model</text
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>P(fork handles|acoustic data): posterior probability</text
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>P(acoustic data): marginal likelihood</text
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>P(fork handles): prior probability</text
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>P(acoustic data|fork handles): likelihood</text
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/></g
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>Model</text
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>P(four candles|acoustic data): posterior probability</text
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/></g
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>P(acoustic data): marginal likelihood</text
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>P(four candles): prior probability</text
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/></g
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>P(acoustic data|four candles): likelihood</text
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><g fill="gray" text-rendering="optimizeSpeed" font-family="sans-serif" transform="translate(147,46)" stroke="gray"
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>Evidences</text
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/></g
><g text-rendering="geometricPrecision" shape-rendering="crispEdges" font-family="sans-serif" transform="translate(242,44)"
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>Acoustic data</text
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