Conditioning and Intervening

We consider the dispute between causal decision theorists and evidential decision theorists over Newcomb-like problems. We introduce a framework relating causation and directed graphs developed by Spirtes et al. (1993) and evaluate several arguments in this context. We argue that much of the debate between the two camps is misplaced; the disputes turn on the distinction between conditioning on an event E as against conditioning on an event I which is an action to bring about E. We give the essential machinery for calculating the effect of an intervention and consider recent work which extends the basic account given here to the case where causal knowledge is incomplete. 1 The Markov condition 2 Intervening and conditioning 3 Inquiry and decision Two essays have come to our attention. One, a recent piece in Science, claims that despite the expense of 200 million dollars the Agency for Health Care Policy and Research (AHCPR) has failed to extract from records of medical data a single correct conclusion relevant to decisions to alter medical practice. The other, an essay of Robert Nozick's long familiar to most philosophers [1969], concerns difficulties in the theory of rational decision-making that arise when actions and their outcomes are both influenced by some feature of the world not under the agent's control. We will argue that both essays turn on the same logical facts, and that partial resolutions of the problems to which they call attention are found in recent work on causal inference and the design of empirical studies. Our exposition invokes a parallelism in the philosophical discussion of rational decision-making and the statistical discussion of experimental design. We begin with Nozick's problem. Suppose you believe, truly and with justification, that a genetic factor causes people both to smoke and to contract cancer of the lung, and that smoking itself has no influence on disease. Suppose further that you believe you would enjoy smoking but it is very much more important to you that you not die of cancer. The probabilities you assign accord with these beliefs: the probability that you will get cancer, given that you smoke (in 1 Science [1994], 263, pp. 1080-2. by gest on F ebuary 9, 2011 bjps.oxjournals.org D ow nladed fom 1002 C. Meek and C. Glymour the future) is greater than the corresponding probability given that you do not smoke (in the future), and cancer and smoking are independent in probability conditional on the value of your genotype. Your utilities likewise accord with these preferences. A simple calculation shows the expected utility of not smoking is greater than the expected utility of smoking; but, whatever your unknown genotype, you are better off smoking than not. Ought you to smoke? Nozick's essay considers variants of this question, which he views as indications of a conflict between two decision theoretic principles: if, in every possible circumstance, one action gives outcomes at least as good as another and in some circumstances better, then the first action should be preferred to the second (weak dominance), and if the sum over all possible states of the world of the probability of the state multiplied by the value of the outcome produced by one action in that state is greater than the like sum for a second action, then the first action is to be preferred to the second (expected utility). Nozick's conclusion is that the expected utility principle must give way to the dominance principle 'if the actions or decisions to do the actions do not affect, help bring about, influence, and so on, which state obtains .. .' Since Nozick's essay appeared, a considerable literature has developed around 'causal decision theory'. Following a suggestion of Robert Stalnaker, Alan Gibbard and William Harper claimed the issue is not between dominance and expected utility, but rather between two forms of the expected utility principle, one using probabilities of states conditional on actions and the other using probabilities for subjunctive (Gibbard and Harper say 'counterfactual') claims as to the consequences were some particular action to be taken. A related theory has been published by David Lewis. Avoiding subjunctives, Brian Skyrms proposed that one should choose the action that maximizes the sum, over each outcome of interest and each 'maximally specific specification of factors outside our influence', of the product of the probability of the outcome given the action and the factor value, the probability of the factor value, and the utility of the outcome, action, and factor value. Where no unique set of such specific factors is known, Skyrms proposed that the probabilities be mixtures over 2 Newcomb's problem—you must decide either to open only the second of two boxes or to open both, knowing that a perfect predictor of your decision put a thousand dollars in the first box and one million dollars in the second if he predicted you would open only the second box, and nothing in the second box if he predicted you would open both—is formally a special case of a problem of this sort, in which the decision-maker believes that both the probability of smoking and the probability of contracting cancer given the appropriate genotype are one, and strict dominance holds. 3 R. Stalnaker [1972]; A. Gibbard and W. Harper [1978]; D. Lewis [1981]. Stalnaker's suggestion is anticipated by P. Fishburn [1964]. by gest on F ebuary 9, 2011 bjps.oxjournals.org D ow nladed fom Conditioning and Intervening 1003 alternative sets of specific factors, and he showed his theory to be formally equivalent to the account proposed by Lewis. Brad Armendt has given a representation theorem for Skyrms' theory in terms of axioms on preferences over appropriate objects. No idea is without objections. Ellery Eells [1982] argued that in assessing choices the probabilities used must be conditional on the total evidence, and that in the relevant cases Nozick and others consider the disposition to take a particular action is itself evidence conditional on which the state of the world and the action are independent. By using 'evidental' decision theory and claiming the situation contains additional information, Eells obtains, in almost all cases, the same decisions as do advocates of causal decision theory. At least in the most celebrated case of this kind, Newcomb's problem, Teddy Seidenfeld rejects altogether Nozick's conclusion that conditional expected utility must give way to dominance, and rejects the causal decision theorists' calculations: for example, in the extreme case of Newcomb's problem with a perfect predictor, one is faced with a decision under certainty: you are certain to be better off if you open only one box. 1 The Markov condition Most of the discussion of causal decision theory, both for and against, has shared assumptions about the connection between beliefs about causality and beliefs about probability (or about the connection between causality and probability) that are perfectly reflected in the statistical literature on causal inference. Most of the philosophical discussion, from Nozick on, supposes that the causal relations in the smoking example, which we may represent graphically as sA: imply a 'factorization' of the probabilities; in obvious notation: P(S,C,G) = P(S|G)P(C|G)P(G). All of the philosophical commentators appear to agree that if smoking and cancer have no effect on one another and have only genotype as their 4 B. Skyrms [1980], [1982] and [1984]; B. Armendt [1988]. For a view similar to Skyrms, see D. Papineau [1989]. 5 T. Seidenfeld [1990]. 6 There are exceptions. In their [1986], Eells and Sober appeal to the notion 'interactive fork' in which the causal relations are represented as in the graph above but the factorization does not hold. In our view all putative examples of'interactive forks' that are not from quantum mechanics are simply cases where further causal connections, as between S and C or between other causes and S and C, have been omitted. That of course was the point of Simon's [1954]. For a discussion, see Spines, Glymour, and Scheines [1993]. by gest on F ebuary 9, 2011 bjps.oxjournals.org D ow nladed fom 1004 C. Meek and C. Glymour common cause, then they are independent conditional on genotype, or in the notation now common among statisticians: