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Bayesian inference 5
<?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 07:40:06 GMT-0400 (EDT)-->
<XholonWorkbook>
<Notes><![CDATA[
Xholon
------
Title: Bayesian inference 5
Description:
Url: http://www.primordion.com/Xholon/gwt/
InternalName: 9ed00ed60c3750822d49
Keywords:
My Notes
--------
This is associated with XholonWorkbooks 8688253668d5fa489b4f, ceb4083319ebfa4a7bd0 and c48e278ac02dce038398,
and is based on "Example 3: Flipping Coins" from "Bayes' Rule - A Tutorial Introduction to Bayesian Analysis" by Dr James V Stone [1].
Here, our task is to decide how unfair a coin is, based on just two
coin flips. Normally, we assume that coins are fair or unbiased, so that
a large number of coin flips (eg 1000) yields an equal number of heads
and tails. But suppose there was a fault in the machine that minted
coins, so that each coin had more metal on one side or the other, with
the result that each coin is biased to produce more heads than tails, or
vice versa. Specifically, 25% of the coins produced by the machine have
a bias of 0.4, and 75% have a bias of 0.6. By definition, a coin with a
bias of 0.4 produces a head on 40% of flips, whereas a coin with a bias
of 0.6 produces a head on 60% of flips (on average). Now, suppose we
choose one coin at random, and attempt to decide which of the two bias
values it has.
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." -->
<Theta_0.4/> <!-- hypothesis that the coin has a bias of 0.4 (heads on 40% of flips on average) -->
<Theta_0.6/> <!-- hypothesis that the coin has a bias of 0.6 (heads on 60% of flips on average) -->
</HypothesisBE>
<EvidenceBE> <!-- E data "the evidence E corresponds to new data that were not used in computing the prior probability" -->
<HeadsTails/> <!-- evidence from two flips of the same coin x = (x_h,x_t) -->
</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/HeadsTails"/>
<port name="hypothesis" connector=".[@roleName='p(heads tails|theta_0.4)']/../../../HypothesesBE/Theta_0.4"/>
<port name="hypothesis" connector=".[@roleName='p(heads tails|theta_0.6)']/../../../HypothesesBE/Theta_0.6"/>
</LikelihoodBE>
<PriorProbabilityBE>
<port name="hypothesis" connector=".[@roleName='p(theta_0.4)']/../../../HypothesesBE/Theta_0.4"/>
<port name="hypothesis" connector=".[@roleName='p(theta_0.6)']/../../../HypothesesBE/Theta_0.6"/>
</PriorProbabilityBE>
<MarginalLikelihoodBE>
<port name="evidence" connector="../../../EvidencesBE/HeadsTails"/>
</MarginalLikelihoodBE>
<PosteriorProbabilityBE>
<port name="hypothesis" connector=".[@roleName='p(theta_0.4|heads tails)']/../../../HypothesesBE/Theta_0.4"/>
<port name="hypothesis" connector=".[@roleName='p(theta_0.6|heads tails)']/../../../HypothesesBE/Theta_0.6"/>
<port name="evidence" connector="../../../EvidencesBE/HeadsTails"/>
</PosteriorProbabilityBE>
</xholonClassDetails>
<BayesianSystem>
<!-- see Figure 1.7 [1] -->
<HypothesesBE>
<Theta_0.4/>
<Theta_0.6/>
</HypothesesBE>
<EvidencesBE>
<HeadsTails/>
</EvidencesBE>
<ModelsBE>
<!-- the goal is to select the most probable of these candidate models, this is called 'model selection' -->
<ModelBE>
<LikelihoodBE roleName="p(heads tails|theta_0.4)" val="0.24"/>
<PriorProbabilityBE roleName="p(theta_0.4)" val="0.25"/>
<MarginalLikelihoodBE roleName="p(heads tails)" val="0.24"/>
<PosteriorProbabilityBE roleName="p(theta_0.4|heads tails)"/> <!-- calculated result should be 0.25 -->
</ModelBE>
<ModelBE>
<LikelihoodBE roleName="p(heads tails|theta_0.6)" val="0.24"/> <!-- maximum likelihood estimate (MLE) -->
<PriorProbabilityBE roleName="p(theta_0.6)" val="0.75"/>
<MarginalLikelihoodBE roleName="p(heads tails)" val="0.24"/>
<PosteriorProbabilityBE roleName="p(theta_0.6|heads tails)"/> <!-- calculated result should be 0.75 -->
</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(2));
}
}
]]></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(2));
pratio.println("In summary:\n It's about " + (1 / pratio.val()).toFixed(0)
+ " times more probable that the coin is " + pprob2.hypothesis.xhc().name()
+ " rather than " + pprob1.hypothesis.xhc().name() + ".");
}
}
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>Posterior ratio</text
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>Models</text
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>P(theta_0.6|heads tails): posterior probability</text
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>P(heads tails): marginal likelihood</text
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>P(theta_0.6): prior probability</text
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>P(heads tails|theta_0.6): likelihood</text
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>P(theta_0.4): prior probability</text
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>P(heads tails|theta_0.4): likelihood</text
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>Evidences</text
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>Heads tails</text
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>Hypotheses</text
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>Theta_0.6</text
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>Theta_0.4</text
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