Epistemic Processes: A Basis for Statistics and Quantum Theory

176 Pages·2018·1.838 MB·English

by  Inge S. Helland

Inge S. Helland
Epistemic
Processes
A Basis for Statistics and
Quantum Theory
Epistemic Processes
Inge S. Helland
Epistemic Processes
A Basis for Statistics and Quantum Theory
123
IngeS.Helland
DepartmentofMathematics
UniversityofOslo
Oslo,Norway
ISBN978-3-319-95067-9 ISBN978-3-319-95068-6 (eBook)
https://doi.org/10.1007/978-3-319-95068-6
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Preface
Allhumandecisions,includingthosemadeinscientific experimentsandin obser-
vational studies, are made in some context. Such contexts are being explicitly
consideredin thisbook.To do so,a conceptualvariableis definedasanyvariable
whichcanbedefinedbya(groupof)human(s)—forinstancescientists—inagiven
setting. Such variables are classified. Sufficiency and ancillarity in a statistical
model used in scientific investigations are defined conditionally in the scientists’
context. The conditionality principle, the sufficiency principle and the likelihood
principle are generalized, and a rule for when one should not condition on an
ancillary is motivated by examples. Model reduction is discussed in generalfrom
thepointofviewthatthereexistsamathematicalgroupactingupontheparameter
space of the model. It is shown that a natural extension of this whole discussion
gives a conceptual fundament from which the formalism of quantum theory can
be discussed. This can also be considered as an argument for an epistemological
basis for quantum theory, a kind of basis that has also been advocated by part of
the quantum foundation community in recent years. Born’s celebrated formula is
shown to follow from a focused version of the likelihood principle together with
some reasonable assumptions on rationality connected to experimental evidence.
ThequestionsaroundBell’sinequalityareapproachedbyusinganepistemicpoint
of view connected to each observer. The Schrödinger equation is derived from
reasonable assumptions. The objective aspects of the world are identified with
the ideal observational results upon which all real and imagined observers agree.
Philosophicalconclusionsfrommypointofdeparturearediscussed.
Thisisaverybriefsummaryofthisbook.Attheoutset,itiswrittenforseveral
groups of readers: (a) physicists interested in the foundation of their science, (b)
statisticians interested in the foundation of their science and its relationship to
modernphysics,(c)philosophersofscience,(d)studentswithagoodbackgroundin
mathematicsand(e)mathematicianswithaninterestinthefoundationofempirical
science.Tohavesuchadiversityofreadersinmindisquiteachallenge.Somewill
experiencepartofthetextasrathertechnical;otherswillseetheapproachasunusual
comparedtothewaytheyareusedtoseetheirownsciencepresented.Iencourage
the reader to start with a relatively open mind, but also criticism of my way of
v
vi Preface
thinkingiswelcome.Thetextconcentratesonthefoundation;thereareverymany
aspects of both quantum theory and statistical inference that are not covered. On
the foundationallevel, I argue that there is a connectionbetween these two areas.
This may be an unusual way of thinking, especially for physicists, and it implies
a special interpretationofquantummechanics.Asan interpretationitis related to
therecentQuantumBayesianapproach,brieflycalledQBism,butItrytoemphasize
thatBayesianismisabranchofstatisticalinferencetheory,notofprobabilitytheory.
Also, I will allow other approaches to statistical inference. Looking at quantum
mechanics as related to statistical inference will presuppose that there are data.
Thesedataareprovidedbymacroscopicmeasurementapparatuses.Intheordinary
approach towards quantum mechanics, these measurement apparatuses are first
discussedatalaterstage;hereIwillassumefromthebeginningthattheyarethere.
Inadiscussionofabriefversionoftheaccountgiveninthisbook,theQuantum
BayesianRuedigerSchackwrote:“Themaindifferencebetweenyourapproachand
QBism is that your approach is phrased in terms of acquiring information about
variables. QBism on the other hand is concerned with decision theory...”. I will
discussmyrelationtoQBisminmoredetailbelow.Iappreciatethatdecisiontheory
isimportant,buthereIprefertheQuantumDecisionTheorypromotedinaseriesof
papersbyYukalovandSornette;see Sect.1.3below.Also, see myremarksto this
theoryinSect.5.5,whereIjustseedecisionasaprimitiveconcept.Myclaimisthat
atheoryaboutacquiringinformationaboutvariablesisimportantenoughtowarrant
abook.Afterallitsaimistocoverthebasisofstatisticsasascienceandalsovery
much of quantum theory as a science. Some of my more qualitative views on the
processofmakingdecisionsarediscussedinChap.6.
Statisticianstalkaboutdataontheonehandandparametersofstatisticalmodels
ontheotherhand.Here,Iwillgeneralizetheparameterconceptinsuchawaythat
the new concept also covers prediction and other statements about single units. I
callthenewconcepte-variable,anabbreviationforepistemicconceptualvariable.
In a quantum mechanical setting, simple e-variables will correspond to what is
usually is called observables.As a statistician I want to distinguishbetween these
observables and the corresponding data values that result from experiments. A
statistician will call these estimates. However, in quantum theory, this distinction
is not always clear. The reason from a statistical point of view is as follows: The
e-variablesof elementary quantumtheory are discrete, and experiments are often
veryaccurate.Thisresult,inconfidenceintervals,credibilityintervalsorprediction
intervalsaroundthetruevaluethatmaydegeneratetoasinglepoint,whichactually
is the true value. Thus, in such cases the distinction between estimates and true
value is blurred out, and much of statistical theory becomes irrelevant. At the
foundational level, I will nevertheless keep the distinction and insist that this is
relatedtostatisticalinference.Assuch,itismadeinacontext,asalreadymentioned.
Theobserver,andthe informationavailabletotheobserver,playsa crucialrolein
quantummechanics.
I acknowledge that many readers may be confused by this book, especially
readerswhoknowalittle(ormore)oftheusualtreatmentofquantummechanics.I
thinkthattheonlythingIcanofferthesereadersatpresentisaletterthatIwroteto
Preface vii
mycolleagueBarbaraHellerrecently.BarbaraisastatisticianlivinginChicago,and
shehasakeeninterestinquantummechanics.Afterreadingapartofmymanuscript,
shewrotetomethatshewasconfused.Hereismyanswer.
DearBarbara,
Iunderstandverywellyourconfusion.—
What does statistics have to offer, in my opinion? It is true that statistics is
being used to a large extent by experimental physicists, but I have been looking
for somethingmore fundamental.Can one think of quantumtheoryand statistical
inference as having partly a common basis? I think so, and I have tried to sketch
such a basis in my book. The common basis is, as I see it, that of an epistemic
process:aprocesstoachieveknowledge.InthesimplestcasehavingwhatIcallan
e-variable(epistemicconceptualvariable)θ,askingaquestiontonature:whatisθ?
Andobtainingananswerintermsofinformationaboutθ.
In statistics, θ is most often a continuous parameter, and the answer can be
in terms of a confidence interval or a Bayesian credibility interval. In elementary
quantummechanics,θ canbediscrete,andinanidealmeasurementwecanobtain
adefiniteanswer:θ =u.
Furthermore,inquantumtheory,wehavewhatIcallcomplementarye-variables:
thinkof spincomponentsin differentdirections.We can obtaininformationabout
θa andaboutθb,butthevector(θa,θb)isinaccessible.
IntheHilbertspaceformulation,ane-variableisidentifiedwithanoperator,the
possiblevaluesofθ aretheeigenvaluesofthatoperatorandverymanystatevectors
canbeidentifiedbytheeventsofthetypeθ =u.Ihavemadeapointofdiscussing
thisidentificationfromseveralpointsofviewinmybook.
Where do the probabilities in quantum theory come from? They are given by
Born’sformula,aformulaforP(θb = u |θa = u )intermsofthestatevectors.I
k j
have a rather long derivation of Born’s formula in my book, using two premises:
(1) A focused likelihood principle, which can be motivated from the likelihood
principle in statistics, a basic principle which again can be derived from fairly
obviousassumptions:aconditionalityprincipleandasufficiencyprinciple.(2)An
assumptionofperfectrationality.
Of course, this is not the full story, but it appears to me to be a very useful
beginningfor understandingquantum theory from assumptions that are related to
thebasicassumptionsofstatisticaltheory.
I hope that this has clarified my points of view to some extent, and I sincerely
hopethatitmayhavecontributedtoremovesomeofyourconfusion.
*
Thereadersjustinterestedinmyviewsonthefoundationofquantummechanics
mayskip Chap.2, but Sect.2.3 shouldbe lookedat, and the conceptof likelihood
asthejointprobability(probabilitydensity)ofthedata,seenasafunctionofthee-
variable/parameter(Sect.2.1.3),shouldbeunderstood.Section3.2maybedropped
viii Preface
ifthegeneralizedlikelihoodprincipleofSect.3.2.4istakenforgranted.Section4.4
maybedroppedinthefirstreading.Theveryimpatientreadermaygodirectlyfrom
Chap.1toSect.5.15.
Aas,Norway IngeS.Helland
References
Bargmann,V.(1964).NoteonWigner’sTheoremonsymmetryoperations. JournalofMathemat-
icalPhysics,5,862–868.
Bjørnstad,J.F.(1990).Predictivelikelihood:Areview.StatisticalScience,5,242–265.
Gelman,A.,&Robert,C. P.(2013).“Notonlydefendedbutalsoapplied”:Theperceivedabsurdity
ofBayesianinference.TheAmericanStatistician,67,1–5.
Hammond, P. J. (2011). Laboratory Games and Quantum Behavior. The Normal Form with a
SeparableStateSpace.Workingpaper.DepartmentofEconomics,UniversityofWarwick.
Helland,I.S.(2008).Quantummechanicsfromfocusingandsymmetry.FoundationsofPhysics,
38,818–842.
Smolin,L.(2011).Arealensembleinterpretationofquantummechanics.aXiv.1104.2822[quant-
ph].
Wigner,E,P.(1959).Grouptheoryanditsapplicationtothequantummechanicsofatomicspectra.
NewYork:Academic.
Acknowledgements
I am grateful to Philip Goyal for inviting me to the workshop on Reconstructing
Quantum Theory at the Perimeter Institute in 2009 on the basis of Helland
(2008). GudmundHermansen has done some of the calculations in connection to
Example 5.16. Also, thanks to Arne B. Sletsjøe for discussions in effect leading
to Proposition4.2,to ErikAlfsen for givingme the preprintHammond(2011),to
Kingsley Jones for making me aware of Wigner (1959) and of Bargmann (1964),
to Joan Ennis for clarification of the meaning of the term “epistemic”, to Chris
EnnisforsendingmeSmolin(2011),forrecommendingtomethebookbyAnton
Zeilingerandforgeneraldiscussions,toPekkaLahtiforinformationrelatedtothe
Bornrule,toDagNormannforinformationaboutpropositionallogic,toGudmund
Hermansen, Celine Cunen and Emil Aas Stoltenberg for the reference to Gelman
andRobert(2013),toNilsLidHjortformakingmeawareofBjørnstad(1990),to
Barbara Heller for giving her first reactions to an unfinished manuscript, to Bent
Selchau for readingand givingcommentsto Chap.1 and notleast for helpingme
through my own epistemic processes, to Paul Busch for comments on the final
manuscriptandfinallytoHaraldMartensandChrisFuchsfordiscussions.Special
thanksgotoAndreiKhrennikovforthroughouttheyearsarrangingveryinteresting
conferencesonquantumfoundation.BigthanksgotoSharonTubbandtherestof
theFranciscoStreetgangfortheiremotionalandintellectualstimulationswhenever
we meet. Finally, I want to thank my family for everything they have sacrificed
whileIhavebeenseekingafoundationrelatedtomyviewsofscience.
ix
Contents
1 TheEpistemicViewUponScience......................................... 1
1.1 Introduction............................................................ 1
1.2 DifferentViewsontheFoundationofQuantumMechanics......... 4
1.3 TheoryofDecisions:Focusing—Context............................ 7
1.4 ThePBRTheorem.AToyModel..................................... 8
1.5 EpistemicProcesses................................................... 9
1.5.1 E-VariablesinSimpleEpistemicQuestions................ 10
1.5.2 E-VariablesinStatistics...................................... 11
1.5.3 E-VariablesinCausalInference............................. 11
1.5.4 E-VariablesinQuantumMechanics......................... 12
1.5.5 RealandIdealMeasurementsinQuantumMechanics..... 13
1.5.6 QuantumStates,TheirInterpretations,andaLink
totheEnsembleInterpretation .............................. 14
1.5.7 QuantumStatesforSpin1/2Particles ...................... 16
1.5.8 InaccessibleConceptualVariables
andComplementarity........................................ 17
References..................................................................... 18
2 StatisticalInference.......................................................... 21
2.1 BasicStatistics......................................................... 21
2.1.1 Probability.................................................... 21
2.1.2 StatisticalModels............................................ 24
2.1.3 InferenceforContinuousParameters ....................... 25
2.1.4 InferenceforDiscretee-Variables........................... 30
2.2 GroupActionsandModelReduction................................. 32
2.3 Interlude................................................................ 36
References..................................................................... 38
3 InferenceinanEpistemicProcess.......................................... 41
3.1 ConceptualVariablesandContexts................................... 41
3.2 Data;GeneralizedSufficiencyandAncillarity....................... 44
3.2.1 Sufficiency.................................................... 45
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