Classical and quantum causal inference

54
Classical and quantum causal inference (An introduction to techniques and open questions) Sally Shrapnel

Transcript of Classical and quantum causal inference

Page 1: Classical and quantum causal inference

Classical and quantum causal inference

(An introduction to techniques and open questions)

SallyShrapnel

Page 2: Classical and quantum causal inference

Classicalcausalinference

...theLaplacianconceptionismoreintunewithhumanintuitions.Thefewesoteric

quantumexperimentsthatconflictwiththepredictionsoftheLaplacianconception

evokesurpriseanddisbelief,andtheydemandscientistsgiveupdeeplyentrenched

intuitionsaboutlocalityandcausality.Ourobjectiveistopreserve,explicateand

satisfy- notdestroy- thoseintuitions.(Pearl,2009)[26]

Page 3: Classical and quantum causal inference

Classicalcausalinference

Glymour onBellexperiments:

...realexperiments...createassociationsthathavenocausalexplanation

consistentwiththeMarkovassumption,andtheMarkovassumptionmustbe

appliedtoobtainthatconclusion.Youcansaythereisnocausalexplanationof

thephenomenon,orthatthereisacausalexplanationbutitdoesn’tsatisfythe

Markovassumption.(Glymour,2006)[124]

Page 4: Classical and quantum causal inference

Classicalcausalinference

Currentquantumtheoryandexperimentsshowthatthisassumptiondoesnotholdat

thequantumlevel,wheretherearenonlocalvariablesthatappeartohaveadirect

causaleffectonothers.WhilethesecasesdonotimplythatthecausalMarkov

assumptiondoesnothold,theydosuggestthatwemayseemoreviolationsofthis

assumptionatthequantumlevel.However,inpractice,theCausalMarkovassumption

assumptionappearstobeareasonableworkingassumptioninmostmacroscopic

systems.

Page 5: Classical and quantum causal inference

Outline

1. Classicalcausalinference

2. Quantumcausalinference

3. Greatintheory,butwhataboutinpractice?

4. Machinelearningtotherescue?

Page 6: Classical and quantum causal inference

Causalinference

Petersetal,2018

p(Ph,A)

p(Ph,B)

Page 7: Classical and quantum causal inference

Causalstructurematters

Petersetal,2018

Page 8: Classical and quantum causal inference

interventionistcausation

CausationisNOTsimplycorrelation

Correlationsdonotenableustodistinguishbetweeneffectiveandineffectivestrategiesthatbringaboutspecificends(Cartwright,1979)

“interventioninvariantmodel”

Page 9: Classical and quantum causal inference

interventionistcausation

CausationisNOTsimplycorrelation

• correlationsdonotenableustodistinguishbetweeneffectiveandineffectivestrategiesthatbringaboutspecificends(Cartwright,1979)

Goldstandard

Page 10: Classical and quantum causal inference

interventionistcausation

Interveningandrandomisation oftennotpossible

• Lookforfootprintsinthedatatogiveuscluestothetopologyofthecausalstructure

• Needprinciplesorassumptionstogofromjointdistributionoverobservedvariablestocausalstructure.

Page 11: Classical and quantum causal inference

Causalinference

Reichenbach’sprinciple:ifXandYarestatisticallydependent,then

1. thereexistseitheracommoncauseZoradirectcausalrelationshipbetweenXandY,and

2. ZscreensoffXandYfromeachother(X⊥Y⏐ Z)

X

Z

Y

X Y

X Y

Page 12: Classical and quantum causal inference

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)

Page 13: Classical and quantum causal inference

CausalgraphicalmodelCasualmodel=graph+causalparametersCausalparameters=distributionofeachvariableconditionedonitsparents

X4

X3

X5

X1

X2

Page 14: Classical and quantum causal inference

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

Page 15: Classical and quantum causal inference

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

Page 16: Classical and quantum causal inference

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

Page 17: Classical and quantum causal inference

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

Page 18: Classical and quantum causal inference

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)

Page 19: Classical and quantum causal inference

Causalinference

X4

X3=2

X5

X1

X2

N4

N5

N2

N1

Interventionseparatesvariablefromantecedentcauses

𝑋1 = 𝑓1(𝑁1)𝑋2 = 𝑓2(𝑁2)

𝑋3 = 2𝑋4 = 𝑓4(𝑋3, 𝑁4)

Page 20: Classical and quantum causal inference

Whyisitcausal?

LocalCPDaregeneratedbyautonomouscausalmechanisms thatexistbetweenparentsandchildren.

Causalstructureofmodelisisomorphictonetworkofautonomouscausalmechanisms.

Interventions bringlocaldistributionundercontrolofexperimenteranddon’tdisruptothermechanismsinthemodel

Thisstructurehaspragmaticvalue– ittellsushowtoacttobringaboutcertainends

Page 21: Classical and quantum causal inference

Latentvariables?

GraphnotMarkovian andFaithful?

Implicitassumptionthatthisisalwaysaconsequenceofunmeasuredcommoncauses(latentvariables)

Inprinciple,locatingandmeasuringsuchvariablesoughttorestoreMarkovianity tothemodelforsomecausalstructure.

Page 22: Classical and quantum causal inference

Constraintbasedmethods

Algorithms:PC

– Identifiesadjacenciesviadependencies

– IdentifiesV-structuresX YZXZXYZXYZY

– Propagationrulestoorientremainingedgestoavoidcycles

– IdentifiesMarkovequivalentsetofDAGsunderCMA,CFA,acyclicity andCSA

Page 23: Classical and quantum causal inference

Constraintbasedmethods

PROBLEMS

Givesyougraphbutnotthestructuralequations(can’tinfercounterfactuals).

StatisticaltestsacceptorrejectCIatagivenconfidenceintervalandgraphstructureisverysensitivetohowyousetthisconfidencelevel.

CItestingisunjustifiedforfinitedata(forarbitrarilycomplexfunctions)

Propagationrulestendstopropagateerrors.

Whatabouttwovariables?

Page 24: Classical and quantum causal inference

AlternativestoCItesting?

Generalprinciples?

Keyideatoretainisindependenceofcausalmechanisms:

“Statisticalcorrelationsbetweenvariablesinasystemaretheresultofacausalgenerativeprocess thatiscomposedofautonomousmodulesthatdonotinformorinfluenceeachother.”

Forbivariatecasethisreducestoindependenceofcause andmechanismrelatingcausetoeffect.

𝑝 𝑐, 𝑒 = 𝑝 𝑐)𝑝 𝑒 𝑐 (𝑐 → 𝑒

Page 25: Classical and quantum causal inference

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

Page 26: Classical and quantum causal inference

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)

Page 27: Classical and quantum causal inference

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

Page 28: Classical and quantum causal inference

Crowdsourcedcausaldiscovery

Guyon,(2013)

Principledmethods(withvoting)– 0.6accuracyonclassificationtask

Generalised – 0.8(lowlevelfeaturesofjointdistribution- 9000!)

Notlearningthefunctionalrelationships,justdirectionality

Page 29: Classical and quantum causal inference

machinelearningmethods

Data=x Model=f(x)p(y|x)

10

Optimise andregularise!

Label=y

Page 30: Classical and quantum causal inference

machinelearningmethods

Supervisedlearning

Largedatabasesofsampledvariablepairswithknowncause-effectrelation.

Castasclassificationproblem.

Taskistogeneralise solutiontounseendatasets(borrowexistingregularisationmethodsfromML).

BUTneedtofeedit“groundtruth”models~motivatedbycommonsense,domainknowledge.

Page 31: Classical and quantum causal inference

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!

Page 32: Classical and quantum causal inference

Classicalcausalinference

Causalinferencefromjointdistributionsisverydifficult.

Nicetheoreticalandconceptualapproaches,butmanypracticaldifficulties.

Machinelearningseemsapromising(bruteforce)approachbutatthecostofinterpretability(fornow).

Causalinferencetotheexclusionofotherdomainknowledgeisanoccupationalhazard…

Page 33: Classical and quantum causal inference

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.

Page 34: Classical and quantum causal inference

Bellinequalities

2006Glymour:

Belltheoremisaparticularexampleofamoregeneraltheoryofcausalinference:

Causalgraphicalmodels

Glymour,Clark "MarkovPropertiesandQuantumExperiments,"inW.DemopoulosandI.Pitowsky,eds. PhysicalTheoryandItsInterpretation:EssaysinHonorofJeffreyBub,Springer2006.

Page 35: Classical and quantum causal inference

Quantumcausalinference

“AnycausalmodelwhichcanreproduceBell-inequalityviolationswhilerespectingtheobservedindependences…willnecessarilyviolateaprinciplethatisatthecoreofallthebestcausaldiscoveryalgorithms [Faithfulness].”

Page 36: Classical and quantum causal inference

Quantumcausalinference?

r

Non-localityviolatesfaithfulness.

• Settingsaremarginallyindependent𝐴 ⊥ 𝐵

• Nosignalling𝑋 ⊥ 𝐵 𝐴; 𝑌 ⊥ 𝐴 𝐵

Page 37: Classical and quantum causal inference

Quantumcausalinference

r

Retrocausality violatesfaithfulness.

• Settingsaremarginallyindependent𝐴 ⊥ 𝐵

• Nosignalling𝑋 ⊥ 𝐵 𝐴; 𝑌 ⊥ 𝐴 𝐵

Page 38: Classical and quantum causal inference

Quantumcausalinference

r

Superdeterminism violatesfaithfulness.

• Settingsaremarginallyindependent𝐴 ⊥ 𝐵

• Nosignalling𝑋 ⊥ 𝐵 𝐴; 𝑌 ⊥ 𝐴 𝐵

Page 39: Classical and quantum causal inference

Quantumcausalinference

Conclusion:CItestingnotrichenoughmethodologytocapturequantumcausalrelations

Page 40: Classical and quantum causal inference

Threeapproaches

1. ClassicalapproachPlugquantumdataintoclassicalalgorithms.Usefulascertificationforquantumness.Doesitreallyhelpwithcausalexplanation?

2. Quantumdomainisacausal.

3. Tryanddevelopamoregeneralversionofcausalinference.

Page 41: Classical and quantum causal inference

Acausal?

Page 42: Classical and quantum causal inference

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)

Page 43: Classical and quantum causal inference

Desiderata

1. Empirical =Theformalismshouldallowforthediscoveryofcausalstructurefromempiricaldata(causalstructure- canactasanoracleforinterventions).

2. Explanation =All correlationsbetweenempiricallyderiveddatashouldbeaccountedforvianotionsofdirect,indirectorcommoncauserelations,i.e.thereshouldbeno“unexplained”correlations.

3. Classicality=Classicalcausalmodelsshouldberecoveredasalimitingcaseofquantumones.

Page 44: Classical and quantum causal inference

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

=

Page 45: Classical and quantum causal inference

Contextuality?

=

Shrapnel,CostaandMilburn,NJP (2018)ShrapnelandCosta,Quantum (2018)

𝑝 𝑗 = 𝑇𝑟(𝐸J𝜌)

Unique!

Unique!

Page 46: Classical and quantum causal inference

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).

Page 47: Classical and quantum causal inference

Quantumcausalmodels

A

B

C

D

G

F

E

Causalmodelis“interventioninvariant”

Non-Markovian?(unmodelled (latent)commoncause)- extendingthemodeltoincludesuchnodesrestorestheMarkovpropertytothecausalgraph

Canprovethatallclassicalcausalmodelscanbegivenaquantumrepresentation(allinstrumentsfixedinsomebasis)

Page 48: Classical and quantum causal inference

Inprinciple:

MeasurementDataà graphthat

(i) Accountsforallcorrelations(classicalorquantum)viaautonomouscausalmechanisms

(ii) Actsasanoracleforfutureinterventions

Needcausaldiscoveryalgorithms….

Page 49: Classical and quantum causal inference

Discoveryalgorithm:

1. Determinesifprocessiscausallyordered

2. Checksifalllatentvariablesareincluded

3. IfMarkoviangivesuniqueDAG

Giarmatzi andCosta,NQI,2018

Page 50: Classical and quantum causal inference

Stillworktodo

Quantumcausalinferenceiscomputationallyand practicallyveryhard.

Needtoinputprocesswhichrequiresinformationallycompletetomography–exponentialinnumberofvariables.

Needtoknowdimensionofinputsystemsandnumberofsubsystems.

Switch?Indefinitecausalorder?LargerclassesofWthataretheoreticallypossible.

Page 51: Classical and quantum causal inference

Supervisedlearningofprocess

Task=>classifyprocessasMarkovianvsnon-Markovian,=>estimatedimensionofnon-Markovianenvironment.

SimulatedataandusesupervisedMLtechnique– labelledprocesses.

TrainedRandomForestRegressor,testonunseendata(frominsideandoutsideoutsidetrainingrange)

99%accurateonMarkovianvsnon-Markovian.

95%accurateondimensionalityofenvironment.

Nolossofaccuracyorgeneralityonless-than-informationallycompletedata(only20%offeaturesincluded).

Page 52: Classical and quantum causal inference

Supervisedlearningofprocess

Caveats:

Simulateddata– tryexperimentaldatanext.

Scaling?

Transferabletodifferentsizeprocesses?

Interpretability?

Page 53: Classical and quantum causal inference

Conclusions

Causalinferenceisimpossiblewithoutassumptions.

Independenceofcausalmechanisms(includinginterventions)iskey.

Machinelearningmethodsmayprovideapowerfultoolforidentificationofcausalstructure(butatsomecosttointerpretability).

Implicationsof“theoryblind”classical-quantumcausalinference?

Page 54: Classical and quantum causal inference

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)


Machine Learning and Causal Inference

Machine Learning and Causal Inference

Causal Inference with Noisy and Missing Covariates via ...papers.nips.cc/paper/7924-causal-inference-with-noisy...Causal Inference with Noisy and Missing Covariates via Matrix Factorization

Causal Inference with Noisy and Missing Covariates via ...papers.nips.cc/paper/7924-causal-inference-with-noisy...Causal Inference with Noisy and Missing Covariates via Matrix Factorization

Causal Inference and Ambiguous Manipulations

Causal Inference and Ambiguous Manipulations

Causal Inference with Observational Datajustinesarey.com/causal-inference.pdf · 1 What is \causal inference?" As Sekhon1 notes in his essay on causal inference, every student of

Causal Inference with Observational Datajustinesarey.com/causal-inference.pdf · 1 What is \causal inference?" As Sekhon1 notes in his essay on causal inference, every student of

Causal inference

Causal inference

Causal inference - pubdocs.worldbank.orgpubdocs.worldbank.org/.../13-b-Technical-Causal-Inference-Leite-ENG… · Fundamental problem of causal inference o For a given unit u, we

Causal inference - pubdocs.worldbank.orgpubdocs.worldbank.org/.../13-b-Technical-Causal-Inference-Leite-ENG… · Fundamental problem of causal inference o For a given unit u, we

Placebo Tests for Causal Inference

Placebo Tests for Causal Inference

How Big Data and Causal Inference Work Together in Health ... · –Three Ideas for Causal Inference –Ideas that aim at solving the problem of causal inference in observational

How Big Data and Causal Inference Work Together in Health ... · –Three Ideas for Causal Inference –Ideas that aim at solving the problem of causal inference in observational

Module . s: Causal Inference - edX€¦ · CAUSAL INFERENCE AND IMPACT EVALUATION Causal inference and impact evaluation is all about attributing a change in an outcome of interest

Module . s: Causal Inference - edX€¦ · CAUSAL INFERENCE AND IMPACT EVALUATION Causal inference and impact evaluation is all about attributing a change in an outcome of interest

CompSci 295, Causal Inference

CompSci 295, Causal Inference

Bayesian Causal Inference: A Tutorial · Bayesian Causal Inference: A Tutorial Fan Li Department of Statistical Science Duke University ... I Causal inference under the potential

Bayesian Causal Inference: A Tutorial · Bayesian Causal Inference: A Tutorial Fan Li Department of Statistical Science Duke University ... I Causal inference under the potential

Causal inference in practice

Causal inference in practice

Bayesian Causal Inference - uni-muenchen.de...from causal inference have been attracting much interest recently. [HHH18] propose that causal [HHH18] propose that causal inference stands

Bayesian Causal Inference - uni-muenchen.de...from causal inference have been attracting much interest recently. [HHH18] propose that causal [HHH18] propose that causal inference stands

Causal Inference in Education Policy Research · 2020-02-28 · Causal Inference Stapleton, McNeish & Mao Ñ 1 Causal Inference in Education Policy Research MLDS Research Series December

Causal Inference in Education Policy Research · 2020-02-28 · Causal Inference Stapleton, McNeish & Mao Ñ 1 Causal Inference in Education Policy Research MLDS Research Series December

"Classical" Inference

"Classical" Inference

Causal Inference: prediction, explanation,

Causal Inference: prediction, explanation,

Elements of Causal Inference - OAPEN

Elements of Causal Inference - OAPEN

Causal Inference MSM

Causal Inference MSM

Origo: causal inference by compression · 2018. 6. 29. · Origo: causal inference by compression 287 – a practical framework for causal inference based on MDL, – a causal inference

Origo: causal inference by compression · 2018. 6. 29. · Origo: causal inference by compression 287 – a practical framework for causal inference based on MDL, – a causal inference

Mediation: The Causal Inference Approach

Mediation: The Causal Inference Approach