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Classical and quantum causal inference
(An introduction to techniques and open questions)
SallyShrapnel
Classicalcausalinference
...theLaplacianconceptionismoreintunewithhumanintuitions.Thefewesoteric
quantumexperimentsthatconflictwiththepredictionsoftheLaplacianconception
evokesurpriseanddisbelief,andtheydemandscientistsgiveupdeeplyentrenched
intuitionsaboutlocalityandcausality.Ourobjectiveistopreserve,explicateand
satisfy- notdestroy- thoseintuitions.(Pearl,2009)[26]
Classicalcausalinference
Glymour onBellexperiments:
...realexperiments...createassociationsthathavenocausalexplanation
consistentwiththeMarkovassumption,andtheMarkovassumptionmustbe
appliedtoobtainthatconclusion.Youcansaythereisnocausalexplanationof
thephenomenon,orthatthereisacausalexplanationbutitdoesn’tsatisfythe
Markovassumption.(Glymour,2006)[124]
Classicalcausalinference
Currentquantumtheoryandexperimentsshowthatthisassumptiondoesnotholdat
thequantumlevel,wheretherearenonlocalvariablesthatappeartohaveadirect
causaleffectonothers.WhilethesecasesdonotimplythatthecausalMarkov
assumptiondoesnothold,theydosuggestthatwemayseemoreviolationsofthis
assumptionatthequantumlevel.However,inpractice,theCausalMarkovassumption
assumptionappearstobeareasonableworkingassumptioninmostmacroscopic
systems.
Outline
1. Classicalcausalinference
2. Quantumcausalinference
3. Greatintheory,butwhataboutinpractice?
4. Machinelearningtotherescue?
interventionistcausation
CausationisNOTsimplycorrelation
Correlationsdonotenableustodistinguishbetweeneffectiveandineffectivestrategiesthatbringaboutspecificends(Cartwright,1979)
“interventioninvariantmodel”
interventionistcausation
CausationisNOTsimplycorrelation
• correlationsdonotenableustodistinguishbetweeneffectiveandineffectivestrategiesthatbringaboutspecificends(Cartwright,1979)
Goldstandard
interventionistcausation
Interveningandrandomisation oftennotpossible
• Lookforfootprintsinthedatatogiveuscluestothetopologyofthecausalstructure
• Needprinciplesorassumptionstogofromjointdistributionoverobservedvariablestocausalstructure.
Causalinference
Reichenbach’sprinciple:ifXandYarestatisticallydependent,then
1. thereexistseitheracommoncauseZoradirectcausalrelationshipbetweenXandY,and
2. ZscreensoffXandYfromeachother(X⊥Y⏐ Z)
X
Z
Y
X Y
X Y
Causalinference
Jointdistributionisthesameinallthreecases:causalstructure capturedbydifferentpossiblefactorisations
X
Z
Y
X Y
X Y
! 𝑝 𝑋 𝑍 𝑝 𝑌 𝑍 𝑝(𝑍)�
)
𝑝(𝑌)𝑝(𝑋|𝑌)
𝑝(𝑋)𝑝(𝑌|𝑋)
𝑝(𝑋, 𝑌)
CausalgraphicalmodelCasualmodel=graph+causalparametersCausalparameters=distributionofeachvariableconditionedonitsparents
X4
X3
X5
X1
X2
Structuralcausalmodel
𝑋1 = 𝑓1(𝑁1)𝑋2 = 𝑓2(𝑁2)
𝑋3 = 𝑓3(𝑋1, 𝑁3)𝑋4 = 𝑓4(𝑋3, 𝑁4)
𝑋5 = 𝑓5 𝑋3, 𝑋2, 𝑁5 X4
X3
X5
X1
X2
N4
N5
N2N3
N1
MarkovCondition
X4
X3
X5
X1
X2
N4
N5
N2N3
N1
parentsNon-descendants
Graphically,children areconditionallyindependentoftheirnon-descendants,giventheirparents.
(𝑋 ⊥ 𝑌 𝑍 𝐺→ (𝑋 ⊥ 𝑌 𝑍 𝑃
FaithfulnessCondition
X4
X3
X5
X1
X2
N4
N5
N2N3
N1
parentsNon-descendants
Graphically,childrenareconditionallyindependentoftheirnon-descendants,giventheirparents.
Independenciesfoundinthedistributionareonly thoseimpliedbytheMarkovcondition
(𝑋 ⊥ 𝑌 𝑍 𝐺 → (𝑋 ⊥ 𝑌 𝑍 𝑃
(𝑋 ⊥ 𝑌 𝑍 𝐺 ← (𝑋 ⊥ 𝑌 𝑍 𝑃
FaithfulnessCondition
X4
X3
X5
X1
X2
N4
N5
N2N3
N1
parentsNon-descendants
Graphically,childrenareconditionallyindependentoftheirnon-descendants,giventheirparents.
Independenciesfoundinthedistributionareonly thoseimpliedbytheMarkovcondition
(𝑋 ⊥ 𝑌 𝑍 𝐺 → (𝑋 ⊥ 𝑌 𝑍 𝑃
(𝑋 ⊥ 𝑌 𝑍 𝐺 ← (𝑋 ⊥ 𝑌 𝑍 𝑃
Interventions
X4
X3
X5
X1
X2
N4
N5
N2
N1
Interventionseparatesvariablefromantecedentcauses
𝑋1 = 𝑓1(𝑁1)𝑋2 = 𝑓2(𝑁2)
𝑋3 = 𝑓3(X1,N3)𝑋4 = 𝑓4(𝑋3, 𝑁4)
Causalinference
X4
X3=2
X5
X1
X2
N4
N5
N2
N1
Interventionseparatesvariablefromantecedentcauses
𝑋1 = 𝑓1(𝑁1)𝑋2 = 𝑓2(𝑁2)
𝑋3 = 2𝑋4 = 𝑓4(𝑋3, 𝑁4)
Whyisitcausal?
LocalCPDaregeneratedbyautonomouscausalmechanisms thatexistbetweenparentsandchildren.
Causalstructureofmodelisisomorphictonetworkofautonomouscausalmechanisms.
Interventions bringlocaldistributionundercontrolofexperimenteranddon’tdisruptothermechanismsinthemodel
Thisstructurehaspragmaticvalue– ittellsushowtoacttobringaboutcertainends
Latentvariables?
GraphnotMarkovian andFaithful?
Implicitassumptionthatthisisalwaysaconsequenceofunmeasuredcommoncauses(latentvariables)
Inprinciple,locatingandmeasuringsuchvariablesoughttorestoreMarkovianity tothemodelforsomecausalstructure.
Constraintbasedmethods
Algorithms:PC
– Identifiesadjacenciesviadependencies
– IdentifiesV-structuresX YZXZXYZXYZY
– Propagationrulestoorientremainingedgestoavoidcycles
– IdentifiesMarkovequivalentsetofDAGsunderCMA,CFA,acyclicity andCSA
Constraintbasedmethods
PROBLEMS
Givesyougraphbutnotthestructuralequations(can’tinfercounterfactuals).
StatisticaltestsacceptorrejectCIatagivenconfidenceintervalandgraphstructureisverysensitivetohowyousetthisconfidencelevel.
CItestingisunjustifiedforfinitedata(forarbitrarilycomplexfunctions)
Propagationrulestendstopropagateerrors.
Whatabouttwovariables?
AlternativestoCItesting?
Generalprinciples?
Keyideatoretainisindependenceofcausalmechanisms:
“Statisticalcorrelationsbetweenvariablesinasystemaretheresultofacausalgenerativeprocess thatiscomposedofautonomousmodulesthatdonotinformorinfluenceeachother.”
Forbivariatecasethisreducestoindependenceofcause andmechanismrelatingcausetoeffect.
𝑝 𝑐, 𝑒 = 𝑝 𝑐)𝑝 𝑒 𝑐 (𝑐 → 𝑒
Asymmetryofcauseandeffect
𝑝 𝑥, 𝑦 = 𝑝 𝑥)𝑝 𝑦 𝑥 (𝑥 → 𝑦
𝑝 𝑥, 𝑦 = 𝑝 𝑦)𝑝 𝑥 𝑦 (𝑦 → 𝑥
𝑦 = 𝑥@ + 𝑥 + 𝑁
Wantlowstructuralvariabilityofmechanismfordifferentinputvaluesofcause
Alternatives?
Makeassumptionsaboutdatageneratingcausalmechanisms:
Bivariate
1. Additivenoisemodel(ANM)• assumenonlinearfunction,additivenoise,noiseindependentof
cause• Admitsonlyasingleunidentifiablecase:linearfunctionwithgaussian
causeandnoise
2. Post-nonlinearmodel(PNL)• Addsanextranonlinearfunctionwhichisinvertible• Morenon-identifiablecases
3. GaussianProcessInferencemodel(GPI)
4. Algorithmiccomplexity– exploitasymmetriesinfactorisation accordingtoKolmogorovcomplexitymeasures(selectssimplestexplanation)
Alternatives?
Multivariate:
1. ConstraintbasedmethodsPC,FCI(relaxesCS),RFCI.Allrequiredata++
2. Scorebasedalgorithmssearchmodelspaceandminimise aglobalscore• GESexploresgraphspaceusingoperators“addedge”,“removeedge”,
“reverseedge”andoptimises accordingtoBIC.• FGESmorecomputationallyefficient
3. Hybridalgorithms=constraint+scorebased• Max-minHillclimbing(MMHC)buildsskeletonusingCIteststhen
performsagreedyhill-climbingsearchtoorientedges• GFCIusesFGEStosketchgraphandthenCItoorientedges
4.Exploitasymmetrybetweencauseandeffect(LinGAM)+MORE• linearfunctions,non-gaussian sourcenodes,additivenoise
Crowdsourcedcausaldiscovery
Guyon,(2013)
Principledmethods(withvoting)– 0.6accuracyonclassificationtask
Generalised – 0.8(lowlevelfeaturesofjointdistribution- 9000!)
Notlearningthefunctionalrelationships,justdirectionality
machinelearningmethods
Supervisedlearning
Largedatabasesofsampledvariablepairswithknowncause-effectrelation.
Castasclassificationproblem.
Taskistogeneralise solutiontounseendatasets(borrowexistingregularisationmethodsfromML).
BUTneedtofeedit“groundtruth”models~motivatedbycommonsense,domainknowledge.
machinelearningmethods
Generativemodels((CiGAN,CausalGAN,CGNN)
SAM“StructuralAgnosticModel”withpenalised adversariallearning.
Recoverscasualgraphfromdata(learnsbothjointandinterventionaldistributions).
RelaxesCMC,CFC,CSA,acyclicity.
Scalestohundredsofvariables.
Estimatesbothstructureofgraphandfunctionalcausalmechanisms.
90%accurate.
Needtofeedit“groundtruth”models– sensitiveto“Bayeserror”.
Needsverification!
Classicalcausalinference
Causalinferencefromjointdistributionsisverydifficult.
Nicetheoreticalandconceptualapproaches,butmanypracticaldifficulties.
Machinelearningseemsapromising(bruteforce)approachbutatthecostofinterpretability(fornow).
Causalinferencetotheexclusionofotherdomainknowledgeisanoccupationalhazard…
Parachuteusetopreventdeathandmajortraumarelatedtogravitationalchallenge:systematicreviewofrandomised controlledtrialsBMJ 2003; 327
Objectives Todeterminewhetherparachutesareeffectiveinpreventingmajortraumarelatedtogravitationalchallenge.Design Systematicreviewofrandomised controlledtrials.Datasources:Medline,WebofScience,Embase,andtheCochraneLibrarydatabases;appropriateinternetsitesandcitationlists.Studyselection: Studiesshowingtheeffectsofusingaparachuteduringfreefall.Mainoutcomemeasure Deathormajortrauma,definedasaninjuryseverityscore>15.ResultsWewereunabletoidentifyanyrandomised controlledtrialsofparachuteintervention.
Conclusions Aswithmanyinterventionsintendedtopreventillhealth,theeffectivenessofparachuteshasnotbeensubjectedtorigorousevaluationbyusingrandomised controlledtrials.Advocatesofevidencebasedmedicinehavecriticisedtheadoptionofinterventionsevaluatedbyusingonlyobservationaldata.Wethinkthateveryonemightbenefitifthemostradicalprotagonistsofevidencebasedmedicineorganised andparticipatedinadoubleblind,randomised,placebocontrolled,crossovertrialoftheparachute.
Bellinequalities
2006Glymour:
Belltheoremisaparticularexampleofamoregeneraltheoryofcausalinference:
Causalgraphicalmodels
Glymour,Clark "MarkovPropertiesandQuantumExperiments,"inW.DemopoulosandI.Pitowsky,eds. PhysicalTheoryandItsInterpretation:EssaysinHonorofJeffreyBub,Springer2006.
Quantumcausalinference
“AnycausalmodelwhichcanreproduceBell-inequalityviolationswhilerespectingtheobservedindependences…willnecessarilyviolateaprinciplethatisatthecoreofallthebestcausaldiscoveryalgorithms [Faithfulness].”
Quantumcausalinference?
r
Non-localityviolatesfaithfulness.
• Settingsaremarginallyindependent𝐴 ⊥ 𝐵
• Nosignalling𝑋 ⊥ 𝐵 𝐴; 𝑌 ⊥ 𝐴 𝐵
Quantumcausalinference
r
Retrocausality violatesfaithfulness.
• Settingsaremarginallyindependent𝐴 ⊥ 𝐵
• Nosignalling𝑋 ⊥ 𝐵 𝐴; 𝑌 ⊥ 𝐴 𝐵
Quantumcausalinference
r
Superdeterminism violatesfaithfulness.
• Settingsaremarginallyindependent𝐴 ⊥ 𝐵
• Nosignalling𝑋 ⊥ 𝐵 𝐴; 𝑌 ⊥ 𝐴 𝐵
Threeapproaches
1. ClassicalapproachPlugquantumdataintoclassicalalgorithms.Usefulascertificationforquantumness.Doesitreallyhelpwithcausalexplanation?
2. Quantumdomainisacausal.
3. Tryanddevelopamoregeneralversionofcausalinference.
Alternativeapproach
Assumequantumcausalstructureisprimitive,buildacausaltheoryfromthegroundup(usingmathematicalobjectsfromQM),thatrecoversclassicalcausalstructureinasuitablelimit.
Writedefinitionsaccordingtothewayphysicistsuse quantumtheorytomakeinterventionistinferencesthatdistinguishbetweeneffectiveandineffectivestrategies(inmostgeneralform).
Obeysindependenceassumptionsthatunderpinclassicalcausalinference.
CostaandShrapnel(2016)“Quantumcausalmodelling”,NJP,18,063062
Shrapnel(2016)“Usinginterventionstodiscoverquantumcausalstructure”PhDthesis,http://espace.library.uq.edu.au/view/UQ:411093
Shrapnel(2017)“DiscoveringQuantumCausalModels,”TheBritishJournalforthePhilosophyofScience(advancearticle)
Desiderata
1. Empirical =Theformalismshouldallowforthediscoveryofcausalstructurefromempiricaldata(causalstructure- canactasanoracleforinterventions).
2. Explanation =All correlationsbetweenempiricallyderiveddatashouldbeaccountedforvianotionsofdirect,indirectorcommoncauserelations,i.e.thereshouldbeno“unexplained”correlations.
3. Classicality=Classicalcausalmodelsshouldberecoveredasalimitingcaseofquantumones.
Quantumcausalmodels
A
B
C
D
G
F
E
Variables=regions
Values=CP maps
Intervention=instruments
CausalMechanisms=CPTP
Causalstructure=process
Generalised circuit
Processisgeneratedbyautonomouscausalmechanisms(channels).
Mechanisms=deterministicunitaries withunmodelled noise
=
Contextuality?
=
Shrapnel,CostaandMilburn,NJP (2018)ShrapnelandCosta,Quantum (2018)
𝑝 𝑗 = 𝑇𝑟(𝐸J𝜌)
Unique!
Unique!
Quantumcausalmodels
A
B
C
D
G
F
E
Variables=regions
Values=CPmaps
Intervention=instruments
CausalMechanisms=CPTP
Causalstructure=process
=
Assumptions:independenceofinterventionsandmechanisms.Do-calculusisignoreincomingstateandre-prepareoutgoingstate.Markovianity – setoflinearconstraintsontheprocess.Faithfulness– nofine-tunedmechanisms(measuretheoreticsense).
Quantumcausalmodels
A
B
C
D
G
F
E
Causalmodelis“interventioninvariant”
Non-Markovian?(unmodelled (latent)commoncause)- extendingthemodeltoincludesuchnodesrestorestheMarkovpropertytothecausalgraph
Canprovethatallclassicalcausalmodelscanbegivenaquantumrepresentation(allinstrumentsfixedinsomebasis)
Inprinciple:
MeasurementDataà graphthat
(i) Accountsforallcorrelations(classicalorquantum)viaautonomouscausalmechanisms
(ii) Actsasanoracleforfutureinterventions
Needcausaldiscoveryalgorithms….
Discoveryalgorithm:
1. Determinesifprocessiscausallyordered
2. Checksifalllatentvariablesareincluded
3. IfMarkoviangivesuniqueDAG
Giarmatzi andCosta,NQI,2018
Stillworktodo
Quantumcausalinferenceiscomputationallyand practicallyveryhard.
Needtoinputprocesswhichrequiresinformationallycompletetomography–exponentialinnumberofvariables.
Needtoknowdimensionofinputsystemsandnumberofsubsystems.
Switch?Indefinitecausalorder?LargerclassesofWthataretheoreticallypossible.
Supervisedlearningofprocess
Task=>classifyprocessasMarkovianvsnon-Markovian,=>estimatedimensionofnon-Markovianenvironment.
SimulatedataandusesupervisedMLtechnique– labelledprocesses.
TrainedRandomForestRegressor,testonunseendata(frominsideandoutsideoutsidetrainingrange)
99%accurateonMarkovianvsnon-Markovian.
95%accurateondimensionalityofenvironment.
Nolossofaccuracyorgeneralityonless-than-informationallycompletedata(only20%offeaturesincluded).
Supervisedlearningofprocess
Caveats:
Simulateddata– tryexperimentaldatanext.
Scaling?
Transferabletodifferentsizeprocesses?
Interpretability?
Conclusions
Causalinferenceisimpossiblewithoutassumptions.
Independenceofcausalmechanisms(includinginterventions)iskey.
Machinelearningmethodsmayprovideapowerfultoolforidentificationofcausalstructure(butatsomecosttointerpretability).
Implicationsof“theoryblind”classical-quantumcausalinference?
References
Acknowledgements:
Causalparents?Hardy,Brukner,Costa,Oreshkov,Spekkens,Liefer,Chiribella,Tucci,Ried,Cavalcanti,Lal,Henson,Pusey,Chaves,Pienaar,Giarmatzi…..+manymore(Lloyd)Causalsisters?Reidetal.,Allenetal.,Causalchildren?Schmid,Pienaar....+more
Books:“Elementsofcausaldiscovery”,Petersetal.,(2018);“DeepLearning”Goodfellowetal.,(2016)