Causality, Conditional Independence, and Graphical Separation in Settable Systems

We study the connections between causal relations and conditional independence within the settable systems extension of the Pearl causal model (PCM). Our analysis clearly distinguishes between causal notions and probabilistic notions, and it does not formally rely on graphical representations. As a foundation, we provide definitions in terms of suitable functional dependence for direct causality and for indirect and total causality via and exclusive of a set of variables. Based on these foundations, we provide causal and stochastic conditions formally characterizing conditional dependence among random vectors of interest in structural systems by stating and proving the conditional Reichenbach principle of common cause, obtaining the classical Reichenbach principle as a corollary. We apply the conditional Reichenbach principle to show that the useful tools of d-separation and D-separation can be employed to establish conditional independence within suitably restricted settable systems analogous to Markovian PCMs.

[1]  H. White,et al.  Robustness checks and robustness tests in applied economics , 2014 .

[2]  Ross D. Shachter,et al.  Decision-Theoretic Foundations for Causal Reasoning , 1995, J. Artif. Intell. Res..

[3]  David J. Spiegelhalter,et al.  Local computations with probabilities on graphical structures and their application to expert systems , 1990 .

[4]  James J. Heckman,et al.  1. The Scientific Model of Causality , 2005 .

[5]  Haiqing Xu,et al.  Causal Discourse in a Game of Incomplete Information , 2014 .

[6]  J. Neveu,et al.  Mathematical foundations of the calculus of probability , 1965 .

[7]  L. Elton,et al.  THE DIRECTION OF TIME , 1978 .

[8]  S. Lauritzen,et al.  Chain graph models and their causal interpretations , 2002 .

[9]  Halbert White,et al.  Settable Systems: An Extension of Pearl's Causal Model with Optimization, Equilibrium, and Learning , 2009, J. Mach. Learn. Res..

[10]  J. Pearl,et al.  Logical and Algorithmic Properties of Conditional Independence and Graphical Models , 1993 .

[11]  H. White,et al.  An Extended Class of Instrumental Variables for the Estimation of Causal Effects , 2011 .

[12]  Nancy Cartwright.  Measuring Causes Invariance, Modularity and the Causal Markov Condition , 2000 .

[13]  A. Dawid.  Conditional Independence in Statistical Theory , 1979 .

[14]  R. H. Strotz,et al.  RECURSIVE VS. NONRECURSIVE SYSTEMS: AN ATTEMPT AT SYNTHESIS (PART I OF A TRIPTYCH ON CAUSAL CHAIN SYSTEMS) , 1960 .

[15]  J. Pearl.  Causal diagrams for empirical research , 1995 .

[16]  Mark W. Schmidt,et al.  Causal learning without DAGs , 2008, NIPS Causality: Objectives and Assessment.

[17]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[18]  P. Spirtes,et al.  Causation, prediction, and search , 1993 .

[19]  Wolfgang Spohn,et al.  Stochastic independence, causal independence, and shieldability , 1980, J. Philos. Log..

[20]  Judea Pearl,et al.  Causal networks: semantics and expressiveness , 2013, UAI.

[21]  A. Philip Dawid,et al.  Beware of the DAG! , 2008, NIPS Causality: Objectives and Assessment.

[22]  Steffen L. Lauritzen,et al.  Independence properties of directed markov fields , 1990, Networks.

[23]  James M. Robins,et al.  Testing Edges by Truncations , 2009, IJCAI.

[24]  Dan Geiger,et al.  Identifying independence in bayesian networks , 1990, Networks.

[25]  J. Q. Smith.  Influence Diagrams for Statistical Modelling , 1989 .

[26]  A. Philip Dawid,et al.  Direct and Indirect Effects of Sequential Treatments , 2006, UAI.

[27]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .

[28]  D. Rubin.  Direct and Indirect Causal Effects via Potential Outcomes * , 2004 .

[29]  Judea Pearl,et al.  Complete Identification Methods for the Causal Hierarchy , 2008, J. Mach. Learn. Res..

[30]  N. Wermuth,et al.  Joint response graphs and separation induced by triangular systems , 2004 .

[31]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[32]  H. White,et al.  Testing a conditional form of exogeneity , 2010 .

[33]  Milan Studený.  Formal Properties of Conditional Independence in Different Calculi of AI , 1993, ECSQARU.

[34]  Judea Pearl,et al.  Direct and Indirect Effects , 2001, UAI.

[35]  Chen Avin,et al.  Identifiability of Path-Specific Effects , 2005, IJCAI.

[36]  H. White,et al.  Identification and Identification Failure for Treatment Effects Using Structural Systems , 2013 .

[37]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

[38]  Xun Lu,et al.  Linking Granger Causality and the Pearl Causal Model with Settable Systems , 2009, NIPS Mini-Symposium on Causality in Time Series.

[39]  J. Woodward,et al.  Independence, Invariance and the Causal Markov Condition , 1999, The British Journal for the Philosophy of Science.

[40]  A. P. Dawid,et al.  Corrigenda: Influence Diagrams for Causal Modelling and Inference , 2002 .

[41]  C. Glymour,et al.  STATISTICS AND CAUSAL INFERENCE , 1985 .

[42]  S. Geneletti.  Identifying direct and indirect effects in a non‐counterfactual framework , 2007 .

[43]  J. Robins,et al.  Identifiability and Exchangeability for Direct and Indirect Effects , 1992, Epidemiology.

[44]  Halbert White,et al.  Causal Diagrams for Treatment Effect Estimation with Application to Efficient Covariate Selection , 2011, Review of Economics and Statistics.