Mol evol jc.151216-amb.bayesianstats-teaser

BayesianStatistics

MadeSimple

ÁlvaroMartínezBarrio,PhD

[email protected]/in/ambarrio@ambarrio

!Uppsala,Dec16th2015

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Think Bayes

Bayesian Statistics Made Simple

Version 1.0.3

Allen B. Downey

Green Tea PressNeedham, Massachusetts

Notation:Probability

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• p(A):theprobabilitythatAoccurs!

• p(A|B):theprobabilitythatAoccurs,giventhatBhasoccurred!

• p(AandB)=p(A)p(B|A):Conjointprobability

Thecookieproblem

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Supposetherearetwobowlsofcookies.Bowl1contains30vanillacookiesand10chocolatecookies.Bowl2contains20ofeach.Nowsupposeyouchooseoneofthebowlsatrandomand,withoutlooking,selectacookieatrandom.Thecookieisvanilla.Whatisthe

probabilitythatitcamefromBowl1?

Thecookieproblem

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p(B1|V)=p(B1)p(V|B1)/p(V)

Thecookieproblem

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p(B1|V)=p(B1)p(V|B1)/p(V)

p(B1|V)=(1/2)(3/4)/5/8

History:Bayes’Theorem

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ThomasBayes,

(b.1702,London-d.1761,

TunbridgeWells,Kent)

Intheearly18thcentury,themathematiciansofthetimeknewhowtofindtheprobabilitythat,say,4peopleaged50dieinagivenyearoutofasampleof60iftheprobabilityofanyoneofthemdyingwasknown.

Buttheydidnotknowhowtofindtheprobabilityofone50-yearolddyingbasedontheobservationthat4haddiedoutof60.

History:Bayes’Theorem

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ThomasBayes,

(b.1702,London-d.1761,


Intheearly18thcentury,themathematiciansofthetimeknewhowtofindtheprobabilitythat,say,4peopleaged50dieinagivenyearoutofasampleof60iftheprobabilityofanyoneofthemdyingwasknown.

Buttheydidnotknowhowtofindtheprobabilityofone50-yearolddyingbasedontheobservationthat4haddiedoutof60.

thequestionofinverseprobability

The“diachronic”interpretation

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ThomasBayes,

(b.1702,London-d.1761,


p(H|D)=p(H)p(D|H)/p(D)

• p(H)istheprobabilityofthehypothesisbeforeweseethedata,calledthepriorprobability,orjustprior.

• p(H|D)iswhatwewanttocompute,theprobabilityofthehypothesisa\erweseethedata,calledtheposterior.

• p(D|H)istheprobabilityofthedataunderthehypothesis,calledthelikelihood.

• p(D)istheprobabilityofthedataunderanyhypothesis,calledthenormalizingconstant.

History:Syllogism

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4thcenturyBC

!!

• Majorpremise!!

• Minorpremise!!

• Conclusion

Arhetoricalsyllogism(a3-partdeductiveargument)usedinoratorialpractice.

History:Syllogism

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4thcenturyBC

• Majorpremise:“Allhumansaremortal”!!

• Minorpremise:“AllGreeksarehuman”!

• Conclusion:“AllGreeksaremortal”

History:Syllogism

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4thcenturyBC

• Majorpremise:“Allmortalsdie”!!

• Minorpremise:“Allmenaremortals”!

• Conclusion:“Allmendie”

History:Enthymeme

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4thcenturyBC

!• “Socratesismortalbecausehe’s

human”

!• Majorpremise(unstated):“Allhumansaremortal.”!

• Minorpremise(stated):“Socratesishuman.”!

• Conclusion(stated):“Therefore,Socratesismortal.”

History:Enthymeme

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4thcenturyBC

!• "Heisill,sincehehasacough.”

!!

• “Sinceshehasachild,shehas

givenbirth."

History:Enthymeme

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4thcenturyBC

• Hestartedtoproposethatenthymemesarebasedonprobabilities(eikos),examples,tekmêria(i.e.,proofs,evidences),andsigns(sêmeia).

History:Enthymeme

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4thcenturyBC

• CarolPosterarguesthatenthymemesastruncatedsyllogismswasinventedbyBritishrhetoricians(suchasRichardWhately)intheXVIIIcentury.

Poster,Carol(2003)."Theology,Canonicity,andAbbreviatedEnthymemes".RhetoricSocietyQuarterly33(1):67–103.

Chapter1:Bayes’Theorem

• Mutuallyexclusive:Atmostonehypothesisinthesetcanbetrue!

• Collectivelyexhaustive:Therearenootherpossibilities;atleastoneofthehypotheseshastobetrue




p(D)=p(B1)p(D|B1)+p(B2)p(D|B2)





p(D)=(1/2)(3/4)+(1/2)(1/2)=5/8





p(D)=(1/2)(3/4)+(1/2)(1/2)=5/8

Ifp(A|B)ishardtocompute,orhardtomeasureexperimentally,checkwhetherit

mightbeeasiertocomputetheothertermsinBayes’stheorem,p(B|A),p(A)andp(B).

Chapter2:ComputationalStatistics

• Distribution:


• Distribution:setofvaluesandtheircorrespondingprobabilities.


• Distribution:setofvaluesandtheircorrespondingprobabilities.• Probabilitymassfunction:waytorepresentadistributionmathematically.



• Whentalkingaboutprobabilities,youneedtonormalise(theyshouldaddupto1)




• Thisdistribution,whichcontainsthepriorsforeachhypothesis,iscalled(waitforit)thepriordistribution.

• Toupdatethedistributionbasedonnewdata(avanillacookie!),wemultiplyeachpriorbythecorrespondinglikelihood.

• Thedistributionisnolongernormalized,youneedtorenormalize• Theresultisadistributionthatcontainstheposteriorprobabilityforeachhypothesis,whichiscalled(waitagain!)theposteriordistribution.




• Thisdistribution,whichcontainsthepriorsforeachhypothesis,iscalled(waitforit)thepriordistribution.

• Toupdatethedistributionbasedonnewdata(avanillacookie!),wemultiplyeachpriorbythecorrespondinglikelihood.

• Thedistributionisnolongernormalized,youneedtorenormalize• Theresultisadistributionthatcontainstheposteriorprobabilityforeachhypothesis,whichiscalled(waitagain!)theposteriordistribution.

Terminologyanddesignpatternsofpythonprogramsthatyoucanuseduringtherest

ofthecourse

Chapter3:Estimation

SupposeIhaveaboxofdicethatcontainsa4-sideddie,a6-sideddie,an8-sideddie,a12-sideddie,anda20-sideddie.IfyouhaveeverplayedDungeons&Dragons,youknowwhatIamtalkingabout.SupposeIselectadiefromtheboxatrandom,rollit,andgeta6.WhatistheprobabilitythatIrolledeachdie?

Chapter3:Estimation

SupposeIhaveaboxofdicethatcontainsa4-sideddie,a6-sideddie,an8-sideddie,a12-sideddie,anda20-sideddie.IfyouhaveeverplayedDungeons&Dragons,youknowwhatIamtalkingabout.SupposeIselectadiefromtheboxatrandom,rollit,andgeta6.WhatistheprobabilitythatIrolledeachdie?

Letmesuggestathree-stepstrategyforapproachingaproblemlikethis:1.Choosearepresentationforthehypotheses. 2.Choosearepresentationforthedata. 3.Writethelikelihoodfunction.

Chapter3:Estimation

Mosteller’sFiftyChallengingProblemsinProbabilitywithSolutions

“Arailroadnumbersitslocomotivesinorder1..N.Onedayyouseealocomotivewiththenumber60.Estimatehowmanylocomotives

therailroadhas.”

Chapter3:Estimation

PartIofStatisticalInference.


therailroadhas.”

Chapter3:Estimation

PartIofStatisticalInference.


therailroadhas.”

Therearetwowaystoproceed:!

•Getmoredata. •Getmorebackgroundinformation.

Chapter3:Estimation

• Credibleinterval:Forintervalsweusuallyreporttwovaluescomputedsothatthereisa90%chancethattheunknownvaluefallsbetweenthem(oranyotherprobability).

• Thewidthofthisintervalsuggestshowuncertainweareabouttheconclusionbasedinourunknownvalue.

• Therearetwoapproachestochoosingpriordistributions:• i)informative:bestrepresentsbackgroundinformation• ii)uninformative:intendedtobeasunrestrictedaspossible

Chapter3:Estimation

• Credibleinterval:Forintervalsweusuallyreporttwovaluescomputedsothatthereisa90%chancethattheunknownvaluefallsbetweenthem(oranyotherprobability).

• Thewidthofthisintervalsuggestshowuncertainweareabouttheconclusionbasedinourunknownvalue.

• Therearetwoapproachestochoosingpriordistributions:• i)informative:bestrepresentsbackgroundinformation• ii)uninformative:intendedtobeasunrestrictedaspossible

Inrealworldyouhavetwowaystoproceed:!

Ifyouhavealotofdata,thechoiceofthepriordoesn’tmatterverymuch;informativeanduninformativepriorsyieldalmostthesameresults.

!Ifyoudon’thavemuchdata,usingrelevantbackgroundinformationmakesabig

difference.

DifferencesbetweenBayesiansandNon-Bayesians

AccordingtoJeffGill(CenterforAppliedStatistics,WashU)



ACCP 37th Annual Meeting, Philadelphia, PA [2]

Differences Between Bayesians and Non-BayesiansAccording to my friend Jeff Gill

Typical Bayesian Typical Non-BayesianTypicalBayesian




Typical Bayesian Typical Non-BayesianTypicalBayesian



Typical Bayesian Typical Non-BayesianTypicalNon-Bayesian


Conclusions

• Importanceofmodelling• Followadiscreteapproach:correctfirst,andexpandlater

GeneralApproach

1.Startwithsimplemodelsandimplementtheminclear,readableanddemonstrablycorrectcode.Focusshouldbeongoodmodellingdecisions,notoptimisation

GeneralApproach

1.Startwithsimplemodelsandimplementtheminclear,readableanddemonstrablycorrectcode.Focusshouldbeongoodmodellingdecisions,notoptimisation2.Identifythebiggestsourcesoferror.Perhapsincreasethenumberofvaluesinadiscreteapproximation,increasethenumberofiterationsinaMCsimulation,oradddetailstothemodel

GeneralApproach

1.Startwithsimplemodelsandimplementtheminclear,readableanddemonstrablycorrectcode.Focusshouldbeongoodmodellingdecisions,notoptimisation2.Identifythebiggestsourcesoferror.Perhapsincreasethenumberofvaluesinadiscreteapproximation,increasethenumberofiterationsinaMCsimulation,oradddetailstothemodel3.Isperformancegood?Ifnot,tryoptimisingthen

REFERENCES

• PyContutorials(byAllenDowney)https://sites.google.com/site/simplebayes/!• “ProbablyOverthinkingIt”(byAllenDowney)http://allendowney.blogspot.se/!

• MarkA.Beaumont&BruceRannala(2004)NatureRevGenetics

MonumenttomembersoftheBayesandCottonfamilies,includingThomasBayesandhisfatherJoshua,inBunhillFieldsburialground

Mol evol jc.151216-amb.bayesianstats-teaser

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