Bayesianism and Scientific Reasoning PDF
Preview Bayesianism and Scientific Reasoning
S c This Element explores the Bayesian approach to the logic and h epistemology of scientific reasoning. Section 1 introduces the u p probability calculus as an appealing generalization of classical b logic for uncertain reasoning. Section 2 explores some of the a c vast terrain of Bayesian epistemology. Three epistemological h postulates suggested by Thomas Bayes in his seminal philosophy of Science work guide the exploration. This section discusses modern developments and defenses of these postulates as well as some important criticisms and complications that lie in wait for the Bayesian epistemologist. Section 3 applies the formal tools and principles of the first two sections to a handful of topics in the epistemology of scientific reasoning: confirmation, explanatory b bayesianism reasoning, evidential diversity and robustness analysis, a y hypothesis competition, and Ockham’s Razor. e s ia n and Scientific is m a n d S Reasoning c ie n t about the Series Series Editor ifi c This series of Elements in Philosophy of Jacob Stegenga R e Science provides an extensive overview University of a s of the themes, topics and debates which Cambridge on Jonah N. Schupbach constitute the philosophy of science. in g Distinguished specialists provide an up-to-date summary of the results of current research on their topics, as well as offering their own take on those topics and drawing original conclusions. Cover image: BSIP / Contributor/ Universal Images Group/Getty Images Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to thIeS CSNam 2b51ri7d-g7e2 7C3o (roen tleinrem)s of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 ISSN 2517-7265 (print) Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 Elements in the Philosophy of Science editedby Jacob Stegenga UniversityofCambridge BAYESIANISM AND SCIENTIFIC REASONING Jonah N. Schupbach University of Utah Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108714013 DOI:10.1017/9781108657563 ©JonahN.Schupbach2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-71401-3Paperback ISSN2517-7273(online) ISSN2517-7265(print) CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 Bayesianism and Scientific Reasoning ElementsinthePhilosophyofScience DOI:10.1017/9781108657563 Firstpublishedonline:January2022 JonahN.Schupbach UniversityofUtah Authorforcorrespondence:JonahN.Schupbach, [email protected] Abstract:ThisElementexplorestheBayesianapproachtothelogicand epistemologyofscientificreasoning.Section1introducestheprobability calculusasanappealinggeneralizationofclassicallogicforuncertain reasoning.Section2exploressomeofthevastterrainofBayesian epistemology.ThreeepistemologicalpostulatessuggestedbyThomas Bayesinhisseminalworkguidetheexploration.Thissectiondiscusses moderndevelopmentsanddefensesofthesepostulatesaswellassome importantcriticismsandcomplicationsthatlieinwaitfortheBayesian epistemologist.Section3appliestheformaltoolsandprinciplesofthefirst twosectionstoahandfuloftopicsintheepistemologyofscientific reasoning:confirmation,explanatoryreasoning,evidentialdiversityand robustnessanalysis,hypothesiscompetition,andOckham’sRazor. Keywords:Bayesianism,explanatoryreasoning,formalepistemology, inductivelogic,scientificreasoning ©JonahN.Schupbach2022 ISBNs:9781108714013(PB),9781108657563(OC) ISSNs:2517-7273(online),2517-7265(print) Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 Contents Introduction 1 1 Probability Theory, a Logic of Consistency 4 2 Bayesian Epistemology 34 3 Scientific Reasoning 67 References 108 Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 BayesianismandScientificReasoning 1 Introduction ThisElementintroducesandcriticallyexplorestheBayesianapproachtothe logicandepistemologyofscientificreasoning.Thisapproachisdistinguished byitsconceptualmediumandformalmethod.Bayesianspaintinthemedium ofwhathasvariouslybeencalled“degreeofbelief,”“credence,”“confidence,” andsoon–whichistosaythatBayesianaccountsessentiallyinvolveagra- dational notion of the doxastic attitude that agents have toward propositions. Somecommonterminologynotwithstanding(e.g.,“partialbelief”and“degree ofbelief”),thisconceptisnotastraightforward,gradationalversionofbelief, whichwouldbereferringtosomethinglikeproportionofoutrightbelief.Such anotionwould“maxout”inthecaseofbeliefsimpliciter,butBayesianstendto beveryquickindismissingtheidentificationofmaximaldegreeofbelief,cre- dence,andsoonwithqualitativebelief(Maher,1993;Leitgeb,2013;Buchak, 2013).Muchmorecommonly,theBayesian’sgradationalmediumisdescribed asmaxingoutinthespecialcaseofcertainty(e.g.,Ramsey1926,§3;Jaynes 2003;Jeffrey2004;Leitgeb2013;SprengerandHartmann2019,p.26).Cer- tainty is a doxastic, propositional attitude of agents generalizing naturally to lessextremeattitudesofuncertaintyorconfidence.Accordingly,inanattempt to avoid confusion in this Element, I use these latter terms when referring to theconceptatthecoreofBayesianaccounts.1 Regardingmethod,Bayesiansconducttheirinvestigationsusingthemathe- maticaltoolsofprobability.ThisformalapproachenablesBayesianstopursue theirphilosophicalworkwithrigorandprecision.Theuseofprobabilitytheory goeshandinhandwithBayesianism’sfocusonconfidence;indeed,Bayesians viewtheprobabilitycalculusasanaptformaltoolbecauseoftheiremphasis onconfidenceanduncertainty.Thebridgebetweenmediumandmethodhere istheBayesian’sepistemicinterpretationofprobabilitiesasdegreesof(more orlessoperationalizedand/oridealizedconceptionsof)confidence. Termsofartinprobabilitytheory–like“probable”and“likely”–areoften usedineverydaylanguagetocommunicateepistemicjudgmentsofuncertainty. However,the Bayesian interpretation is far from uncontroversial. Probability theory,mathematicallyspeaking,isabranchofmeasuretheory,completewith an axiomatization and consequent structure (Kolmogorov, 1933). While the epistemic interpretation of probability as a guide to rational confidence was therefromthebeginningofthismath’sdevelopment,muchofthisdevelopment was driven by more objective, physical, and aleatory concepts, applications, 1 Theuseof“partialbelief”and“degreeofbelief”torefertoBayesianism’scentralnotioncan indeedleadtoconfusionandcriticismsofBayesianismthatarerathertooeasy.Forexample, seeHorgan’s(2017,p.236)“conceptualconfusion”objectiontoBayesianism. Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 2 PhilosophyofScience and interpretations (see Hacking 2006, ch. 2). The mathematics of probabil- ity was at best developed only in part as an explication of confidence. Thus, a case needs to be made for the validity and usefulness of the Bayesian’s interpretation. This Element explores the Bayesian approach to logic and epistemology in three parts. Section 1 provides a primer on the elementary mathematics ofprobability,motivatedandpresentedthroughaBayesianlens.Wedevelop probability theory as a conservative generalization of classical logic, which is more readily applicable to reasoning under uncertainty. Probability theory provides a logic of consistency for attitudes of confidence. That is, probabil- itytheorydescribeshowanagent’sconfidences(ataparticulartime)oughtto relateinorderforthemtobeinternallyconsistent.Additionally,thissectiondis- cusses the Bayesian interpretation of probability as a measure of confidence, and it critically evaluates some of the common arguments presented for and againstthisinterpretation. Section 2 explores a number of contentious principles put forward and debated by Bayesians. Unlike the rules discussed in Section 1, these prin- ciples are more characteristically and recognizably epistemological. This is becausetheygobeyondthelogicofconsistencyintheorizingabouthowour confidencesoughttobe,notjustrelatedtoeachother,butsensitivetoandcon- strained by our experience of the world. For direction in our exploration, we turntothepatronsaintofBayesianepistemology,thegoodReverendThomas Bayes,andhisseminal“EssayTowardsSolvingaProblemintheDoctrineof Chances”(1763).Bayessuggeststhreeepistemologicalprinciplesinthiswork, each of which is still much discussed, developed, and debated by contempo- raryBayesianepistemologists.Wediscusstheseprinciplesinsomedetailalong withsomecorrespondingcriticismsandcomplicationsthatlieinwaitforthe Bayesianepistemologist. The final Section 3 displays the potential fruitfulness of the Bayesian approachforthestudyofscientificreasoning.Weintroducejustahandfulof themanytopicsBayesianshavediscussedintheepistemologyofscience:spe- cifically,wefocusontheepistemologyofconfirmation,explanatoryreasoning, evidentialdiversityandrobustnessanalysis,hypothesiscompetition,andOck- ham’s Razor. Via our discussions of these topics, we aim to show that our understandingofsomeimportantconcepts,methods,andstrategiescommonly usedinscientificpracticecanbeimprovedbytakingaBayesianapproach. ForthesakeofkeepingthisworkElement-length,I’veoftenhadtorefrain fromdelvingintointerestingissuesandobviousquestionsleftstandingbymy presentations.Atonelevel,thisnecessityhasbotheredme,sinceitopensme uptocriticismsthatImighthavetriedtopreemptandlimitsmetosimplistic Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 BayesianismandScientificReasoning 3 overviewsofcertaintopics.However,atanotherlevel,Ithinkandhopethatthe resultismoreconducivetopromotingfurtherdiscussion(beitinclassrooms orresearchsettings),andindeedtoinspiringawidervarietyoffutureresearch ontherelevantissues. IfirstlearnedaboutBayesianismthroughaseriesofcoursesItookfromTim- othyMcGrewasanMAstudentatWesternMichiganUniversity(from2004 to 2006). To this day, I owe Tim an enormous debt of gratitude, not only for introducingmetoanenjoyableandfascinatingfieldofstudy,butalsoforbeing suchapatient,caring,andcarefulinstructor.The“dartboardrepresentations” IemployinSection1tracetheirrootstosimilardiagramsTimdevelopedand usedwhenintroducingmetothefield.Mydeepestthanksadditionallygoout toseveralcolleaguesandstudentswhowillinglyandgraciouslygavetheirtime toreaddraftsofthisworkandprovidemewithfeedback.JonathanLivengood andJoshuaBarthulyinparticulareachcarefullyreadandprovidedsubstantial feedbackoncompletedraftsofthisElement;thisworkismuchbetterbecause oftheirgracioushelp.Otherreaderswhogavemeinvaluablefeedbackonpor- tionsofthetextincludeJeanBerroa,LiamEgan,SamuelFletcher,Konstantin Genin,DavidH.Glass,Qining(Tim)Guo,RobertHartzell,DanielMalinsky, ConorMayo-Wilson,LydiaMcGrew,JacobStegenga,MichaelTitelbaum,Jon Williamson, and two anonymous reviewers for Cambridge University Press. Thematerialfor§3.3.2andalargepartof§3.4wasdevelopedincollaboration withDavidH.Glass.MyheartfeltgratitudealsogoesouttoAbbotSilouanand all of the monks at the Monastery of the Holy Archangel Michael (Cañones, NewMexico) forproviding me with gracioushospitality,good conversation, andtheunimaginablypeacefulenvironmentoftheirguesthouse,whereIwrote thebulkofthisElement. While working on this Element, I was supported by the Charles H. Mon- son Mid-Career Award administered by the University of Utah’s Philosophy Departmentandbygrant#61115fromtheJohnTempletonFoundation(which Ico-directedwithDavidH.Glass).TheopinionsexpressedinthisElementare thoseoftheauthoranddonotnecessarilyreflecttheviewsofeitherofthese fundingsources. Finally, “thank you” is not enough when I try to imagine how I should respond to the support, love, and joy that I receive everyday from my wife and children. I don’t deserve such blessings. Truth be told, my confidence is notveryhighthatanyoneofthemwilleverreadtheentiretyofthisElement. Nonetheless,itwouldn’thavehappenedwereitnotforthem,andIdedicateit tothem. Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563 4 PhilosophyofScience 1 Probability Theory, a Logic of Consistency Bayesianismischaracterizedbyitsfocusonthenotionofconfidence(“degree of belief,” “credence,” “partial belief,” etc.) and its corresponding epistemic interpretationofprobabilitytheory.2Bayesiansapplytheprobabilitycalculus, so interpreted, to a wide variety of epistemological principles, concepts, and puzzles.Thissectionintroducesthemathematicsofprobabilitytheorythrough aBayesianlens.Byfirsttouchingonsomefeaturesofclassical,deductivelogic, we highlight the need to generalize this formal logic in order to deal directly with the inevitable uncertainties of scientific (and everyday) inferences. The probabilitycalculus,whengivenaBayesianinterpretation,providesapartic- ularlyappealinggeneralization,resultinginacompellinglogicofconsistency fortheconfidencesofuncertainagents.Wefinishthissectionbyconsidering someargumentsforandagainstthiswayofconstruingprobabilitytheory. 1.1 Logicand Uncertainty Logic is the science of inference. This discipline systematically studies the relation by which conclusions follow from premises. As such, while logic is fundamental to epistemology, the two disciplines are distinct. Epistemol- ogy would surely be lacking if it gave no role to inference in the theory of knowledge,butthereareotherfactorsbesidesinferencethatplaycrucialepis- temologicalroles.3Logicisalsodistinctfromanotherscienceofinference,the psychologyofinference,whichobserves,predicts,andmodelstheinferential behavior of humans actually drawing conclusions from premises.4 Logic, by contrast,ismoreofatheoreticalscience.Ittheorizesabouttheexistenceand nature of general principles relating conclusions and premises in such a way astolegitimize,guide,andregulatesuchbehavior.Theseprinciples,whichthe renownedlogicianGeorgeBoole(1854)calledthe“lawsofthought,”concern thefundamentalnatureofconsequence,orwhatfollowsfromwhat. 2 BayesianismisoftenadditionallycharacterizedasbeingcommittedtoBayes’sRule,aprinci- plelegislatinghowconfidences(explicatedasprobabilities)shouldevolveoveradiachronic process of learning new information. We postpone our discussion of this principle until Section2. 3 Thereareotherconceptionsoflogic,someofwhichencompassepistemology.Forexample, Ramsey(1926,p.87)viewslogicasconsistingoftwoparts,“formallogic”ortheexplicative logicofconsistency,and“inductivelogic”ortheampliativelogicofdiscoveryandtruth.I acceptRamsey’sdistinctionbutamtryingtokeepterminologytidybyreserving“logic”for Ramsey’snotionofformallogic.WhatRamseycalls“inductivelogic”ispartofepistemology asIcharacterizeit–cf.Jeffreys’s(1939,p.1)remark:“Thetheoryoflearningingeneralisthe branchoflogicknownasepistemology.” 4 The wisdom of distinguishing the logical from the psychological may be questioned, how- ever. Kimhi (2018), for example, argues that the sundering of the two is a recent, Fregean developmentthathashaddetrimentalphilosophicaleffects. Downloaded from https://www.cambridge.org/core. IP address: 89.187.163.162, on 08 Feb 2022 at 15:55:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108657563
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