DOCSLIB.ORG

Causal Inference Without Counterfactuals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Causal Inference Without Counterfactuals In Journal of American Statistical Association, Vol. 95, No. 450, 428-431, June 2000. TECHNICAL REPORT With discussions by D. Cox; G. Casella & S. Schwartz; J. Pearl; J. Robins & S. Greenland; R-269 D. Rubin; G. Shafer; L. Wasserman. June 2000 Comment Judea PEARL 1. BACKGROUND ships among observablevariables remain invariantwhen The fieldof statisticshas seenmany well-meaning cru- the values of those variableschange relative to our imme- sades againstthreats from metaphysics and otherheresy. diate observations.For example,Ohm's law (V = IR) as- In its foundingprospectus of 1834,the Royal Statistical sertsthat the ratiobetween the current(I) and the voltage Societyresolved "to excludecarefully all Opinionsfrom (V) across a resistorremains constant for all values of I, its transactionsand publications-toconfine its attentionincluding yet-unobserved values of I. We usually express rigorouslyto facts."This clause was officiallystruck out thisclaim in a functionor a hypotheticalsentence: "Had the in 1858,when it becameobvious that facts void of the- currentin theresistor been I (insteadof theobserved value orycould not take statistics very far (Annals of theRoyal Io) the voltage would have been V = I(Vo/Io)," know- StatisticalSociety 1934, p. 16). ing perfectlywell thatthere is no way to simultaneously KarlPearson launched his ownmetaphysics "red scare" measure I and Io. (Every mathematicalfunction is inter- aboutcausality in 1911:"Beyond such discarded fundamen- preted hypothetically, and the studyof counterfactualsis talsas 'matter'and 'force' lies still another fetish amidst the merelya studyof standardmathematical functions.) Such inscrutablearcana of modern science, namely, the category sentencesappear to be counterfactual,because they deal of causeand effect" (Pearson 1911, p. iv). Pearson'sobjec- withunobserved quantities that differ from (and henceseem tionto theoreticalconcepts such as "matter"and "force" to contradict)those actually observed. Nonetheless, this cir- was so fierceand his rejectionof determinismso absolute cumstantialnonobservability and apparentcontradiction do thathe consignedstatistics to almosta centuryof neglect notdiminish whatsoever the ability to submitphysical laws withinthe study of causalinference. Philip Dawid was one to empiricaltests. Scientific methods thrive on attemptsto of a handfulof statisticianswho boldly protested the stale- confirmor falsifythe predictionsof such laws. mateover causality: "Causal inference is one of themost The same applies to stochastic processes (or data- important,most subtle, and most neglected of all theprob- generationmodels), usually writtenin the formof func- lemsof statistics"(Dawid 1979). tionalrelations y = f(x, U), whereX and U standfor two In thepast two decades,owing largely to progressin sets of randomvariables, with joint distributionP(x, u), counterfactual,graphical, and structuralanalyses, causal- and f is a function(usually of unknownform) that deter- ityhas beentransformed into a mathematicaltheory with mines the value of the outcome Y = y in termsof ob- well-definedsemantics and well-foundedlogic, and many served and unobservedquantities, X = x and U = u. To practicalproblems that were long regarded as eithermeta- see how counterfactualsand joint probabilitiesof counter- physicalor unmanageable can now be solvedusing elemen- factualsemerge from such a stochasticmodel, I consider a tarymathematics. (See Pearl2000 fora gentleintroduction simple case where Y and X are binaryvariables (e.g., to thecounterfactual, graphical, and structural equation ap- treatmentand response)and U is an arbitrarycomplex set of proachesto causality.)In thearticle, Professor Dawid wel- all othervariables that may influenceY. For any given conditionU the X Y comesthe new progressin causal analysisbut expresses u, relationshipbetween and must be one mistrustof thequasi-deterministic methods by whichthis of the (only) fourbinary functions progresshas beenachieved. fo: y = 0 or {Yo = O,Y,= 0}, Attitudesof towardcounterfactuals and suspicion struc- fl: y = x or {Yo = O,Y1= 1} turalequation models are currently pervasive among statis- x ticians,and Dawid should be commendedfor bringing such f2: y Xor {Yo = 1,Y,= 0}, concernsinto the open. By helpingto dispelmisconcep- and tionsabout counterfactuals, Dawid's article may well have f3: y = or{Yo = 1, Y =1}. (1) rescuedstatistics from another century of stagnationover As u varies along its domain,the only effectit can have causality. on the model is to switchthe relationshipbetween X and Y among these fourfunctions. This partitionsthe domain 2. THE EMPIRICAL CONTENT OF of U into fourequivalence classes, whereeach class con- COUNTERFACTUALS tains those pointsu thatcorrespond to the same function. The word"counterfactual" is a misnomer.Counterfac- The probabilityP(u) thus induces a probabilityfunction tualscarry as clearan empiricalmessage as anyscientific over the potential response pairs {Yo, Y1} shownin (1). This laws,and indeed are fundamentalto them.The essenceof constructionis the inverseof the one discussedin Dawid's any scientificlaw lies in the claimthat certain relation- Section 13; one startswith genuineconcomitants U, and JudeaPearl is Professorof ComputerScience and Statistics,University ? 2000 American Statistical Association of California,Los Angeles,CA 90024 (E-mail:[email protected], Web: Journal of the American Statistical Association www.cs.ucla..edu/-judea/). June 2000, Vol. 95, No. 450, Theory and Methods 428 This content downloaded from 131.179.232.118 on Fri, 12 Sep 2014 18:41:13 PM All use subject to JSTOR Terms and Conditions Pearl: Comment 429 theyturn into jointly distributedcounterfactual concomi- queriesread: tants{Yo, Y1} thatDawid calls metaphysicaland fatalistic. Admittedly,when u standsas the identityof a person, QI = P (Y1 = 0) - P (YO = 0) the mappingof u into the pair {Yo, Y1} appears horridly and fatalistic,as if thatperson is somehowdoomed to reactin = a predeterminedway to treatment(X 1) and no treat- QII = P (YO 1|X = i,Y = 0). (2) ment (X = 0). However,if one views u as the sum total of all experimentalconditions that mightpossibly affect In words,Qll standsfor the probabilitythat my headache thatindividual's reaction (including biological, psychologi- would have stayedhad I not takenaspirin (Yo = 1), given cal, and spiritualfactors, operating both before and afterthe thatI did in fact take aspirin (X = 1) and the headache applicationof the treatment),then the mappingis seen to has gone (Y = 0). (I restrictthe population to personswho evolve reasonablyand naturallyfrom the functionalmodel have headachesprior to consideringaspirin.) Dawid is cor- y = f (x, u). This quasi-deterministicfunctional model mir- rect in statingthat the two queries are of differenttypes, rorsLaplace's conceptionof nature(Laplace 1814), accord- and thelanguage of counterfactualsdisplays this difference ing to whichof nature'slaws are deterministic,and random- and its ramificationsin vividmathematical form. By exam- ness surfacesmerely due to our ignoranceof theunderlying iningtheir respective formulas, one can immediatelydetect boundaryconditions. (The structuralequation models used thatQll is conditionedon the outcomeY = 0, whereasQI in economics,biology, and stochasticcontrol are typical is unconditioned.This impliesthat some knowledgeof the examplesof Laplacian models.)Dawid deteststhis concep- functionalrelationship (between X and Y) mustbe invoked tion. This is not because it ever failed to match macro- in estimatingQll (Balke and Pearl 1994). I challengeDawid scopic empiricaldata (only quantummechanical phenom- to express Qll, let alone formulateconditions for its esti- ena exhibitassociations that might conflict with the Lapla- mationin a counterfactual-freelanguage. For background cian model),but rather because it appearsto standcontrary information,the identificationof QI requires exogeneity to the "familiarstatistical framework and machinery"(Sec. (i.e., randomizedtreatment), whereas thatof Qll requires 7). I fail to see why a frameworkand machinerythat did both exogeneityand monotonicity;both assumptionshave not exactlyexcel in the causal arena shouldbe deprivedof testableimplications (Pearl 2000, p. 294). Epidemiologists enhancementand retooling. are well aware of the differencebetween QI and Qll [they usuallywrite Qll = QI/P (Y = O|X = 1)], thoughthe cor- 3. EMPIRICISM VERSUS IDENTIFIABILITY respondingidentification conditions for Qll are oftennot Dawid's empiricismis summarizedin his abstract: spelled out as clearlyas theycould (Greenlandand Robins 1988). By definition,one can never observe such [counterfactual] quantities,nor assess empiricallythe validityof any model- What is puzzlingin Dawid's articleis thathe considers ing assumptionmade aboutthem, even thoughone's conclu- Qll to be, on one hand,valid and important(Sec. 3) and,on sions may be sensitiveto theseassumptions. the otherhand, untestable (Sec. 11); the two are irreconcil- able. If Qll is valid and important,then one shouldexpect This warningis not entirelyaccurate. Many counter- the magnitudeof Qll to affectsome futuredecisions, and factual modeling assumptionsdo have testable implica- can then use the correctnessof those decisions as a test tions; for example, exogeneity(or ignorability)(Y1 ii X) (henceinterpretation) of theempirical claims made by Qll. > and monotonicity(Y1 (u) Yo(u)) each can be falsifiedby What are those claims,and how can theybe tested? and data comparingexperimental nonexperimental (Pearl Accordingto theinterpretation
Load more

Recommended publications
  • STATISTICAL SCIENCE Volume 36, Number 3 August 2021
    STATISTICAL SCIENCE Volume 36, Number 3 August 2021 Khinchin’s 1929 Paper on Von Mises’ Frequency Theory of Probability . Lukas M. Verburgt 339 A Problem in Forensic Science Highlighting the Differences between the Bayes Factor and LikelihoodRatio...........................Danica M. Ommen and Christopher P. Saunders 344 A Horse Race between the Block Maxima Method and the Peak–over–Threshold Approach ..................................................................Axel Bücher and Chen Zhou 360 A Hybrid Scan Gibbs Sampler for Bayesian Models with Latent Variables ..........................Grant Backlund, James P. Hobert, Yeun Ji Jung and Kshitij Khare 379 Maximum Likelihood Multiple Imputation: Faster Imputations and Consistent Standard ErrorsWithoutPosteriorDraws...............Paul T. von Hippel and Jonathan W. Bartlett 400 RandomMatrixTheoryandItsApplications.............................Alan Julian Izenman 421 The GENIUS Approach to Robust Mendelian Randomization Inference .....................................Eric Tchetgen Tchetgen, BaoLuo Sun and Stefan Walter 443 AGeneralFrameworkfortheAnalysisofAdaptiveExperiments............Ian C. Marschner 465 Statistical Science [ISSN 0883-4237 (print); ISSN 2168-8745 (online)], Volume 36, Number 3, August 2021. Published quarterly by the Institute of Mathematical Statistics, 9760 Smith Road, Waite Hill, Ohio 44094, USA. Periodicals postage paid at Cleveland, Ohio and at additional mailing offices. POSTMASTER: Send address changes to Statistical Science, Institute of Mathematical Statistics, Dues and Subscriptions Office, PO Box 729, Middletown, Maryland 21769, USA. Copyright © 2021 by the Institute of Mathematical Statistics Printed in the United States of America Statistical Science Volume 36, Number 3 (339–492) August 2021 Volume 36 Number 3 August 2021 Khinchin’s 1929 Paper on Von Mises’ Frequency Theory of Probability Lukas M. Verburgt A Problem in Forensic Science Highlighting the Differences between the Bayes Factor and Likelihood Ratio Danica M.
    [Show full text]
  • A Matching Based Theoretical Framework for Estimating Probability of Causation
    A Matching Based Theoretical Framework for Estimating Probability of Causation Tapajit Dey and Audris Mockus Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Abstract The concept of Probability of Causation (PC) is critically important in legal contexts and can help in many other domains. While it has been around since 1986, current operationalizations can obtain only the minimum and maximum values of PC, and do not apply for purely observational data. We present a theoretical framework to estimate the distribution of PC from experimental and from purely observational data. We illustrate additional problems of the existing operationalizations and show how our method can be used to address them. We also provide two illustrative examples of how our method is used and how factors like sample size or rarity of events can influence the distribution of PC. We hope this will make the concept of PC more widely usable in practice. Keywords: Probability of Causation, Causality, Cause of Effect, Matching 1. Introduction Our understanding of the world comes from our knowledge about the cause-effect relationship between various events. However, in most of the real world scenarios, one event is caused by multiple other events, having arXiv:1808.04139v1 [stat.ME] 13 Aug 2018 varied degrees of influence over the first event. There are two major concepts associated with such relationships: Cause of Effect (CoE) and Effect of Cause (EoC) [1, 2]. EoC focuses on the question that if event X is the cause/one of the causes of event Y, then what is the probability of event Y given event Email address: [email protected], [email protected] (Tapajit Dey and Audris Mockus) Preprint submitted to arXiv July 2, 2019 X is observed? CoE, on the other hand, tries to answer the question that if event X is known to be (one of) the cause(s) of event Y, then given we have observed both events X and Y, what is the probability that event X was in fact the cause of event Y? In this paper, we focus on the CoE scenario.
    [Show full text]
  • December 2000
    THE ISBA BULLETIN Vol. 7 No. 4 December 2000 The o±cial bulletin of the International Society for Bayesian Analysis A WORD FROM already lays out all the elements mere statisticians might have THE PRESIDENT of the philosophical position anything to say to them that by Philip Dawid that he was to continue to could possibly be worth ISBA President develop and promote (to a listening to. I recently acted as [email protected] largely uncomprehending an expert witness for the audience) for the rest of his life. defence in a murder appeal, Radical Probabilism He is utterly uncompromising which revolved around a Modern Bayesianism is doing in his rejection of the realist variant of the “Prosecutor’s a wonderful job in an enormous conception that Probability is Fallacy” (the confusion of range of applied activities, somehow “out there in the world”, P (innocencejevidence) with supplying modelling, data and in his pragmatist emphasis P ('evidencejinnocence)). $ analysis and inference on Subjective Probability as Contents procedures to nourish parts that something that can be measured other techniques cannot reach. and regulated by suitable ➤ ISBA Elections and Logo But Bayesianism is far more instruments (betting behaviour, ☛ Page 2 than a bag of tricks for helping or proper scoring rules). other specialists out with their What de Finetti constructed ➤ Interview with Lindley tricky problems – it is a totally was, essentially, a whole new ☛ Page 3 original way of thinking about theory of logic – in the broad ➤ New prizes the world we live in. I was sense of principles for thinking ☛ Page 5 forcibly struck by this when I and learning about how the had to deliver some brief world behaves.
    [Show full text]
  • ALEXANDER PHILIP DAWID Emeritus Professor of Statistics University of Cambridge E-Mail: [email protected] Web
    ALEXANDER PHILIP DAWID Emeritus Professor of Statistics University of Cambridge E-mail: [email protected] Web: http://www.statslab.cam.ac.uk/∼apd EDUCATION 1967{69 University of London (Imperial College; University College) 1963{67 Cambridge University (Trinity Hall; Darwin College) 1956{63 City of London School QUALIFICATIONS 1982 ScD (Cantab.) 1970 MA (Cantab.) 1967 Diploma in Mathematical Statistics (Cantab.: Distinction) 1966 BA (Cantab.): Mathematics (Second Class Honours) 1963 A-levels: Mathematics and Advanced Mathematics (Double Distinction), Physics (Distinction) State Scholarship HONOURS AND AWARDS 2018 Fellow of the Royal Society 2016 Fellow of the International Society for Bayesian Analysis 2015 Honorary Lifetime Member, International Society for Bayesian Analysis 2013 Network Scholar of The MacArthur Foundation Research Network on Law and Neuroscience 2002 DeGroot Prize for a Published Book in Statistical Science 2001 Royal Statistical Society: Guy Medal in Silver 1978 Royal Statistical Society: Guy Medal in Bronze 1977 G. W. Snedecor Award for Best Publication in Biometry EMPLOYMENT ETC. 2013{ Emeritus Professor of Statistics, Cambridge University 2013{ Emeritus Fellow, Darwin College Cambridge 2007{13 Professor of Statistics and Director of the Cambridge Statistics Initiative, Cambridge University 2011{13 Director of Studies in Mathematics, Darwin College Cambridge 2007{13 Professorial Fellow, Darwin College Cambridge 1989{2007 Pearson Professor of Statistics, University College London 1983{93 Head, Department of Statistical Science, University College London 1982{89 Professor of Probability and Statistics, University College London 1981{82 Reader in Probability and Statistics, University College London 1978{81 Professor of Statistics, Head of Statistics Section, and Director of the Statistical Laboratory, Department of Mathematics, The City University, London 1969{78 Lecturer in Statistics, University College London VISITING POSITIONS ETC.
    [Show full text]
  • Theory and Applications of Proper Scoring Rules
    Theory and Applications of Proper Scoring Rules A. Philip Dawid, University of Cambridge and Monica Musio, Università di Cagliari Abstract We give an overview of some uses of proper scoring rules in statistical inference, including frequentist estimation theory and Bayesian model selection with improper priors. 1 Introduction The theory of proper scoring rules (PSRs) originated as an approach to probability forecasting (Dawid 1986), the general enterprise of forming and evaluating probabilistic predictions. The fundamental idea is to develop ways of motivating a forecaster to be honest in the predictions he announces, and of assessing the performance of announced probabilities in the light of the outcomes that eventuate. The early applications of this methodology were to meteorology (Brier 1950) and subjective Bayesianism (Good 1952; de Finetti 1975). However, it is becoming clear that the mathematical theory of PSRs has a wide range of other applications in Statistics generally. The aim of this paper is to give a brief outline of some of these. Some further details will appear in Dawid et al. (2014). After setting out the basic ideas and properties of PSRs in § 2, in § 3 we describe a number of important special cases. Section 4 describes how a PSR supplies an alternative to the likelihood function of a statistical model, and how this can be used to develop new estimators that may have useful properties of robustness and/or computational feasability. In particular, in § 5 we show that, by using an appropriate PSR, it is possible to avoid difficulties associated with intractable normalising constants. In § 6 the same ability to ignore a normalising constant is shown to supply an approach to Bayesian model selection with improper priors.
    [Show full text]
  • PHIL 8670 (Fall, 2015): Philosophy of Statistics
    PHIL 8670 (Fall, 2015): Philosophy of Statistics Samuel C. Fletcher September 12, 2015 Primary Email: scfl[email protected] Office Phone: 612-625-4325 Office Hours: 754 Heller Hall, Tuesdays 3 pm { 4 pm, and by appointment Class Period: 731 Heller Hall, Tuesdays 4 pm { 6 pm Course Description Statistical methods form a significant part of the methodology of every scientific discipline. This seminar will survey a number of central philosophical and conceptual issues in statistics. After a brief review of elementary probability theory and its interpretation, we will consider at length both the classical and Bayesian schools of statistics: their formulations, and arguments pro and con. We will then consider a third school, likelihoodism, that is sometimes considered a compromise between classical and Bayesian methodologies and has gained favor among some philosophers of science. This leads to the likelihood principle, a natural focal point in the debate between these schools on the nature of evidence. The remainder of the seminar will be split between topics on statistical modeling (model verification and model selection), connections between the philosophy of statistics and more traditional topics in the philosophy of science (simplicity, induction and confirmation) and the relation between statistics and causation (randomness, randomization and causal inference). Evaluation Each week, there will be a semi-formal presentation of that week's reading assignment aimed at stimulating a critical discussion. For the first meeting (or two) I will be responsible for presenting, and for each of the remaining meetings, one or two students will be responsible. Students not presenting during a given week are responsible instead for submitting a one- paragraph question about a philosophical/conceptual issue arising in that week's readings to those presenting (and me).
    [Show full text]
  • Computational and Financial Econometrics (CFE 2017)
    CFE-CMStatistics 2017 PROGRAMME AND ABSTRACTS 11th International Conference on Computational and Financial Econometrics (CFE 2017) http://www.cfenetwork.org/CFE2017 and 10th International Conference of the ERCIM (European Research Consortium for Informatics and Mathematics) Working Group on Computational and Methodological Statistics (CMStatistics 2017) http://www.cmstatistics.org/CMStatistics2017 Senate House & Birkbeck University of London, UK 16 – 18 December 2017 ⃝c ECOSTA ECONOMETRICS AND STATISTICS. All rights reserved. I CFE-CMStatistics 2017 ISBN 978-9963-2227-4-2 ⃝c 2017 - ECOSTA ECONOMETRICS AND STATISTICS Technical Editors: Gil Gonzalez-Rodriguez and Marc Hofmann. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any means without the prior permission from the publisher. II ⃝c ECOSTA ECONOMETRICS AND STATISTICS. All rights reserved. CFE-CMStatistics 2017 International Organizing Committee: Ana Colubi, Erricos Kontoghiorghes, Marc Levene, Bernard Rachet, Herman Van Dijk. CFE 2017 Co-chairs: Veronika Czellar, Hashem Pesaran, Mike Pitt and Stefan Sperlich. CFE 2017 Programme Committee: Knut Are Aastveit, Alessandra Amendola, Josu Arteche, Monica Billio, Roberto Casarin, Gianluca Cubadda, Manfred Deistler, Jean-Marie Dufour, Ekkehard Ernst, Jean-David Fermanian, Catherine Forbes, Philip Hans Franses, Marc Hallin, Alain Hecq, David Hendry, Benjamin Holcblat, Jan Jacobs, Degui Li, Alessandra Luati, Richard Luger, J Isaac Miller, Claudio Morana, Bent
    [Show full text]
  • Hyper and Structural Markov Laws for Graphical Models
    Hyper and Structural Markov Laws for Graphical Models Simon Byrne Clare College and Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics This dissertation is submitted for the degree of Doctor of Philosophy University of Cambridge September 2011 Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indi- cated in the text. iii Acknowledgements I would like to thank everyone who helped in the writing of this thesis, especially my supervisor Philip Dawid for the thoughtful discussions and advice over the past three years. I would also like to thank my colleagues in the Statistical Laboratory for their support and friendship. Finally I would like to thank my family, especially Rebecca, without whom I would never have been able to complete this work. v Summary My thesis focuses on the parameterisation and estimation of graphical mod- els, based on the concept of hyper and meta Markov properties. These state that the parameters should exhibit conditional independencies, similar to those on the sample space. When these properties are satisfied, parameter estima- tion may be performed locally, i.e. the estimators for certain subsets of the graph are determined entirely by the data corresponding to the subset. Firstly, I discuss the applications of these properties to the analysis of case-control studies. It has long been established that the maximum like- lihood estimates for the odds-ratio may be found by logistic regression, in other words, the “incorrect” prospective model is equivalent to the correct retrospective model.
    [Show full text]
  • Probability Forecasts and Prediction Markets A
    PROBABILITY FORECASTS AND PREDICTION MARKETS A. Philip Dawid - Julia Mortera ISSN 2279-6916 Working papers (Dipartimento di Economia Università degli studi Roma Tre) (online) Working Paper n° 250, 2019 I Working Papers del Dipartimento di Economia svolgono la funzione di divulgare tempestivamente, in forma definitiva o provvisoria, i risultati di ricerche scientifiche originali. La loro pubblicazione è soggetta all'approvazione del Comitato Scientifico. Per ciascuna pubblicazione vengono soddisfatti gli obblighi previsti dall'art. l del D.L.L. 31.8.1945, n. 660 e successive modifiche. Copie della presente pubblicazione possono essere richieste alla Redazione. esemplare fuori commercio ai sensi della legge 14 aprile 2004 n.106 REDAZIONE: Dipartimento di Economia Università degli Studi Roma Tre Via Silvio D'Amico, 77 - 00145 Roma Tel. 0039-06-57335655 fax 0039-06-57335771 E-mail: [email protected] http://dipeco.uniroma3.it PROBABILITY FORECASTS AND PREDICTION MARKETS A. Philip Dawid - Julia Mortera Comitato Scientifico: Fabrizio De Filippis Francesco Giuli Anna Giunta Paolo Lazzara Loretta Mastroeni Silvia Terzi Probability Forecasts and Prediction Markets Julia Mortera∗ A. Philip Dawidy Abstract We give an overview of various topics tied to the expression of uncertainty about a variable or event by means of a probability distribution. We first consider meth- ods used to evaluate a single probability forecaster and then revisit methods for combining several experts' distributions. We describe the implications of imposing coherence constraints, based on a specific understanding of \expertise". In the final part we revisit from a statistical standpoint some results in the economics literature on prediction markets, where individuals sequentially place bets on the outcome of a future event.
    [Show full text]
  • Extended Conditional Independence and Applications in Causal Inference
    The Annals of Statistics 2017, Vol. 45, No. 6, 1–36 https://doi.org/10.1214/16-AOS1537 © Institute of Mathematical Statistics, 2017 EXTENDED CONDITIONAL INDEPENDENCE AND APPLICATIONS IN CAUSAL INFERENCE BY PANAYIOTA CONSTANTINOU AND A. PHILIP DAWID University of Warwick and University of Cambridge The goal of this paper is to integrate the notions of stochastic conditional independence and variation conditional independence under a more general notion of extended conditional independence. We show that under appropri- ate assumptions the calculus that applies for the two cases separately (axioms of a separoid) still applies for the extended case. These results provide a rig- orous basis for a wide range of statistical concepts, including ancillarity and sufficiency, and, in particular, the Decision Theoretic framework for statistical causality, which uses the language and calculus of conditional independence in order to express causal properties and make causal inferences. 1. Introduction. Conditional independence is a concept that has been widely studied and used in Probability and Statistics. The idea of treating conditional in- dependence as an abstract concept with its own calculus was introduced by Dawid [12], who showed that many results and theorems concerning statistical concepts such as ancillarity, sufficiency, causality, etc., are just applications of general prop- erties of conditional independence—extended to encompass stochastic and non- stochastic variables together. Properties of conditional independence have also been investigated by Spohn [50] in connection with causality, and Pearl and Paz [45], Pearl [44], Geiger et al. [34], Lauritzen et al. [42] in connection with graphi- cal models. For further related theory, see [15–19, 27, 30].
    [Show full text]
  • Sufficient Covariate, Propensity Variable and Doubly Robust
    Sufficient Covariate, Propensity Variable and Doubly Robust Estimation Hui Guo, Philip Dawid and Giovanni Berzuini Abstract Statistical causal inference from observational studies often requires ad- justment for a possibly multi-dimensional variable, where dimension reduction is crucial. The propensity score, first introduced by Rosenbaumand Rubin,is a popular approach to such reduction. We address causal inference within Dawid’s decision- theoretic framework, where it is essential to pay attention to sufficient covariates and their properties. We examine the role of a propensity variable in a normal linear model. We investigate both population-based and sample-based linear regressions, with adjustments for a multivariate covariate and for a propensity variable. In ad- dition, we study the augmented inverse probability weighted estimator, involving a combination of a response model and a propensity model. In a linear regression with homoscedasticity, a propensity variable is proved to provide the same estimated causal effect as multivariate adjustment. An estimated propensity variable may, but need not, yield better precision than the true propensity variable. The augmented inverse probability weighted estimator is doubly robust and can improve precision if the propensity model is correctly specified. Hui Guo Centre for Biostatistics, Institute of Population Health, The University of Manch- arXiv:1501.07761v1 [math.ST] 30 Jan 2015 ester, Jean McFarlane Building, Oxford Road, Manchester M13 9PL, UK, e-mail: [email protected] Philip Dawid Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK, e-mail: [email protected] Giovanni Berzuini Department of Brain and Behavioural Sciences, University of Pavia, Pavia, Italy, e-mail: [email protected] 1 2 Hui Guo, Philip Dawid and Giovanni Berzuini 1 Introduction Causal effects can be identified from well-designed experiments, such as ran- domised controlled trials (RCT), because treatment assignment is entirely unrelated to subjects’ characteristics, both observed and unobserved.
    [Show full text]
  • Decision-Theoretic Foundations for Statistical Causality
    Decision-theoretic foundations for statistical causality A. Philip Dawid∗ April 28, 2020 Abstract We develop a mathematical and interpretative foundation for the enterprise of decision-theoretic statistical causality (DT), which is a straightforward way of representing and addressing causal questions. DT reframes causal inference as \assisted decision-making", and aims to understand when, and how, I can make use of external data, typically observational, to help me solve a decision problem by taking advantage of assumed relationships between the data and my problem. The relationships embodied in any representation of a causal problem require deeper jus- tification, which is necessarily context-dependent. Here we clarify the considerations needed to support applications of the DT methodology. Exchangeability considerations are used to structure the required relationships, and a distinction drawn between intention to treat and intervention to treat forms the basis for the enabling condition of \ignorability". We also show how the DT perspective unifies and sheds light on other popular formalisa- tions of statistical causality, including potential responses and directed acyclic graphs. Key words: directed acyclic graph, exchangeability, extended conditional independence, ignor- ability, potential outcome, single world intervention graph 1 Introduction The decision-theoretic (DT) approach to statistical causality has been described and developed in a series of papers (Dawid 2000; Dawid 2002; Dawid 2003; Didelez et al. 2006; Dawid 2007a; Geneletti 2007; Dawid and Didelez 2008; Dawid and Didelez 2010; Guo and Dawid 2010; Geneletti and Dawid 2011; Dawid 2012; Berzuini et al. 2012b; Dawid and Constantinou 2014; Guo et al. 2016); for general overview see Dawid (2007b); Dawid (2015).
    [Show full text]