In this paper, we develop a formal, systematic approach to sensitivity analysis for arbitrary linear Structural Causal Models SCMs. We start by formalizing sensitivity analysis as a constrained identification problem. We then develop an efficient, graph-based identification algorithm that exploits non-zero constraints on both directed and bidirected edges.

This allows researchers to systematically derive sensitivity curves for a target causal quantity with an arbitrary set of path coefficients and error covariances as sensitivity parameters. Following Halpern , we will call an assignment of values to the exogenous variables a context. In an acyclic SEM, a context uniquely determines the values of all the variables in the model. For instance, if we add the setting. To intervene on a variable is to set the value of that variable by a process that overrides the usual causal structure, without interfering with the causal processes governing the other variables.

More precisely, an intervention on a variable X overrides the normal equation for X , while leaving the other equations unchanged. For example, to intervene on the variable Flame in our example would be to set the flame to a specified level regardless of whether the igniter is pressed or what the gas level is. Perhaps, for example, one could pour kerosene into the grill and light it with a match.

Woodward proposes that we think of an intervention as a causal process that operates independently of the other variables in the model. Randomized controlled trials aim to intervene in this sense.

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For example, a randomized controlled trial to test the efficacy of a drug for hypertension aims to determine whether each subject takes the drug rather than a placebo by a random process such as a coin flip. Factors such as education and health insurance that normally influence whether someone takes the drug no longer play this role for subjects in the trial population.

To represent an intervention on a variable, we replace the equation for that variable with a new equation stating the value to which the variable is set. The new system of equations can then be solved to discover what values the other variables would take as a result of the intervention. In the world described above, our intervention would produce the following set of equations:.

We have struck through the original equation for Flame to show that it is no longer operative. Since the equation connecting Flame to its causes is removed, any changes introduced by setting Flame to 1 will only propagate forward through the model to the descendants of Flame. The intervention changes the values of Flame and Meat cooked , but it does not affect the values of the other variables.

We can represent interventions on multiple variables in the same way, replacing the equations for all of the variables intervened on. Interventions help to give content to the arrows in the corresponding DAG. For example, in our original model, Gas level is a parent of Flame. If we set the value of Igniter to 1 by means of an intervention, and set Gas knob, Gas connected, Meat on, and Meat cooked to any values at all, then intervening on the value of Gas level makes a difference for the value of Flame.

Setting the value of Gas level to 1 would yield a value of 1 for Flame ; setting Gas level to 2 yields a Flame of 2; and so on. A counterfactual is a proposition in the form of a subjunctive conditional. The antecedent posits some circumstance, typically one that is contrary to fact. For example, in our gas grill world, the flame was high, and the meat was well done.

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The antecedent posits a hypothetical state of affairs, and the consequent describes what would have happened in that hypothetical situation. Deterministic SEMs naturally give rise to a logic of counterfactuals. These counterfactuals are called structural counterfactuals or interventionist counterfactuals.

Structural counterfactuals are similar in some ways to what Lewis calls non-backtracking counterfactuals. In a non-backtracking counterfactual, one does not reason backwards from a counterfactual supposition to draw conclusions about the causes of the hypothetical situation.

### 2. Basic Tools

The formalism for representing interventions described in the previous section prevents backtracking from effects to causes. The logic of structural counterfactuals has been developed by Galles and Pearl , Halpern , Briggs , and Zhang a. More precisely, we define well-formed formulas wff s for the language inductively:. In this case, the antecedent is a contradiction, and the counterfactual is considered trivially true. If the antecedent is a disjunction of atomic propositions, or a disjunction of conjunctions of atomic propositions, then the consequent must be true when every possible intervention or set of interventions described by the antecedent is performed.

Consider, for instance,. Hence the counterfactual comes out false. Some negations are treated as disjunctions for this purpose. If the consequent contains a counterfactual, we iterate the procedure. Consider the counterfactual:. In this case, the interventions are performed in a specified order: Flame is first set to 1, and then set to 2.

The differences between structural counterfactuals and Stalnaker-Lewis counterfactuals stem from the following two features of structural counterfactuals:. The truth values of counterfactuals are determined solely by the causal structures of worlds, together with the interventions specified in the their antecedents. No further considerations of similarity play a role. For example, the counterfactual.

These features of structural counterfactuals lead to some unusual properties in the full language developed by Briggs :. To handle the second kind of case, Briggs defines a relation of exact equivalence among Boolean propositions using the state space semantics of Fine Within a world, the state that makes a proposition true is the collection of values of variables that contribute to the truth of the proposition. By contrast, the state that makes.

Propositions are exactly equivalent if they are made true by the same states in all possible worlds. The truth value of a counterfactual is preserved when exactly equivalent propositions are substituted into the antecedent. Briggs provides a sound and complete axiomatization for structural counterfactuals in acyclic SEMs. Many philosophers and legal theorists have been interested in the relation of actual causation. This concerns the assignment of causal responsibility for some event that occurs, based on how events actually play out.

For example, suppose that Billy and Suzy are both holding rocks. Suzy throws her rock at a window, but Billy does not. We can represent this story easily enough with a structural equation model. For variables, we choose:. The equation for W tells us that the window would shatter if either Billy or Suzy throws their rock. The corresponding DAG is shown in Figure 4. But we cannot simply read off the the relation of actual causation from the graph or from the equations.

Note that while it is common to distinguish between singular or token causation, and general or type-level causation see, e. Our causal model does not represent any kind of causal generalization: it represents the actual and possible actions of Billy and Suzy at one particular place and time. Actual causation is not just causal structure of the single case.

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A further criterion for actual causation, defined in terms of the causal structure together with the actual values of the variables, is needed. Following Lewis a , it is natural to try to analyze the relation of actual causation in terms of counterfactual dependence. In our model, the following propositions are all true:. In general, we might attempt to analyze actual causation as follows:. Unfortunately, this simple analysis will not work, for familiar reasons involving preemption and overdetermination.

Here is an illustration of each:. Preemption: Billy decides that he will give Suzy the opportunity to throw first. In fact, Suzy throws her rock, which shatters the window. Billy does not throw. Overdetermination: Billy and Suzy throw their rocks simultaneously. Their rocks hit the window at the same time, shattering it. Either rock by itself would have been sufficient to shatter the window. Much of the work on counterfactual theories of causation since has been devoted to addressing these problems. As an illustration, consider one analysis based closely on a proposal presented in Halpern :.

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In Preemption , let the variables B , S , and W be defined as above. Our context and equations are:. The DAG is shown in Figure 5. Conditions AC 1 and AC 2 are clearly satisfied. In words, this counterfactual says that if neither Billy nor Suzy had thrown their rock, the window would not have shattered. Thus condition AC 3a is satisfied.

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Here is how AC works in this example. These two paths interact in such a way that they cancel each other out, and the value of S makes no net difference to the value of W.

## Linear Causal Modeling with Structural Equations - Dimensions

However, by holding B fixed at its actual value of 0, we eliminate the influence of S on W along that path. The result is that we isolate the contribution that S made to W along the direct path. AC defines actual causation as a particular kind of path-specific effect. To treat Overdetermination , let B , S , and W keep the same meanings. Our setting and equation will be:. The graph is the same as that shown in Figure 4 above. Conditions AC 1 and AC 2 are obviously satisfied. The key idea here is that S is a member of a minimal set of variables that need to be changed in order to change the value of W.

Despite these successes, none of the analyses of actual causation developed so far perfectly captures our pre-theoretic intuitions in every case.