Prediction and Experimental Design with Graphical Causal Models

Abstract

We unify two contemporary theoretical frameworks for representing causal dependencies. Directed graphical models were introduced and developed by Kiiveri, Speed, Wermuth, Lauritzen, Pearl and others. Rubin introduced a framework for analyzing the relation between the conditional probability of Y on X and the distribution Y would have if X were forced to have a particular value. Pratt and Schlaifer have extended Rubin's analysis to offer sufficient counterfactual conditions for the conditional distribution of Y on Z, X= x to equal the conditional distribution of Y on Z when all units in the population are forced to have that value of X. Using two axioms for directed graphical causal models, we obtain rigorous derivations of claims given by Rubin and by Pratt and Schlaifer, and we give general characterizations in terms of causal structure-represented by directed graphs--for Pratt and Schlaifer's notions of the observability of a and the observability of a law with concomitants. Results obtained in the Rubin framework are generalized, and some relevant sampling properties of graphical causal models are obtained. 1 Correspondence: C. Glymour, Department of Philosophy, Carnegie Mellon University, Pittsburgh, Pa. 15213. E-mail cgO9@andrew.cmu.edu 1 Research for this paper was supported in part by contract number N00114-89-J1964 from the Navy Personnel Research and Development Center and the Office of Naval Research. One of the aims of an empirical study may be to predict the effects a general policy would have if put in force, or to predict relevant differences resulting from alternative policies. The interest might be in predicting the differential yield if a field is planted with one species of wheat rather than another; or the difference in number of polio cases per capita if all children are vaccinated against polio as against if none are; or the difference in recidivism rates if parolees are given $600 per month for six months as against if they are given nothing; or the reduction of lung cancer deaths in middle aged smokers if they are given help in quitting cigarette smoking; or the decline in gasoline consumption if an additional dollar tax per gallon is imposed. Such inference problems are puzzling because a policy of treatment creates a potential distribution different from the distribution sampled in observations or experiments, and alternative policies of treatment create alternative potential distributions with alternative statistics. The inference task is to move from a sample of one of these distributions, the one corresponding to passive observation or experimental manipulation, to conclusions about the distribution that would result if a policy were imposed. A further feature makes prediction especially difficult. Empirical studies are often unable to control or randomize all of the relevant variables, with the result that the dependency among variables relevant to prediction may be confounded by unmeasured common causes. In that case, the effect of a policy that manipulates one of the variables cannot be expected to be predictable from sample statistics. There are many examples of predictions whose disappointment may be in part due to confounding. The second Surgeon General's report on smoking and health (1979, p. 43) found that mortality ratios (compared to permanent non-smokers) for those who quit smoking declined with the number of years since quitting, equaling lifelong non-smokers after 15 years. Brownlee (1965), following Fisher (1959) conjectured that such decreases might at least in part be due to self-selection of quitters caused by genetic or cultural factors. The Brownlee hypothesis can be represented by a simple picture.