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Actual causation and the art of modeling

Joseph Y. Halpern, Christopher Hitchcock

arXiv:1106.2652v1cs.AI

TL;DR

Structural equations and choices of variables can shape actual-causation judgments, raising the question of how causal models should be justified. The paper examines structural-equation modeling and adds normality rankings to constrain the contingencies used in counterfactual tests; this approach handles problematic examples, though the authors say more work is needed before causal models are used in applications.

  • Problem

    Structural equations and choices of variables can shape actual-causation judgments, but the paper asks how to justify a model and its choices.

    Source: §1 · p. 3

  • Method

    The paper examines structural-equation models and uses normality rankings to restrict counterfactual contingencies to worlds at least as normal as the actual world.

    Source: §1 · p. 1, §5 · p. 19

  • Results

    The normality-based definition handles problematic examples, including ruling out the bodyguard as a cause of the victim’s survival when the required contingency is less normal than the actual world.

    Source: §5 · p. 19

  • Takeaways & Limitations

    The authors take preliminary steps toward principles for choosing variables and defaults, while concluding that much more work is needed before causal models are used in applications.

    Source: §6 · p. 21

    Adding normality can give modelers enough flexibility to make a cause claim false through their chosen ranking, intensifying the need to justify the model.

    Source: §5 · p. 20

Abstract

from arXiv · show

We look more carefully at the modeling of causality using structural equations. It is clear that the structural equations can have a major impact on the conclusions we draw about causality. In particular, the choice of variables and their values can also have a significant impact on causality. These choices are, to some extent, subjective. We consider what counts as an appropriate choice. More generally, we consider what makes a model an appropriate model, especially if we want to take defaults into account, as was argued is necessary in recent work.

1 Introduction

The paper examines how structural-equation models shape actual-causation judgments and what makes choices of variables, values, and defaults appropriate and defensible. It also reviews how counterfactual dependence and normality-based contingencies address causal cases that simple counterfactual tests mishandle.

1 Introduction

  • Structural equations represent intervention effects, but choices of variables and their values also shape actual-causation judgments and are partly subjective.The authors argue that this makes judgments of actual causation subjective as well, even when people agree on the relevant equations.Source: §1 · p. 2
  • Different experts can give reasonable accounts of an accident by selecting variables that reflect their distinct professional concerns.The paper gives traffic engineering, education, sociology, and psychology as examples of differing causal focuses.Source: §1 · p. 2
  • The paper asks how to justify modeling choices, noting that model stability and interactions between represented events matter.A litigant arguing that faulty brakes caused an accident needs to justify the model, while an opponent needs grounds for challenging it.Source: §1 · pp. 2–3
  • A simple counterfactual-dependence test misses Suzy’s role when Billy’s accurate throw would also have shattered the bottle.The HP definition addresses this by testing dependence under a contingency, such as Billy not throwing.Source: §1 · p. 3
  • Structural-equation accounts can give inappropriate answers in isomorphic cases, motivating proposals to incorporate defaults and normality [Hall 2007; Halpern 2008; Hitchcock 2007a].The proposed approach considers contingencies at least as normal as the actual world, as in attributing lateness to unusually bad traffic rather than a typical departure time.Source: §1 · pp. 3–4
  • Adding defaults makes model justification harder because defaults may represent moral obligations, social conventions, prototypicality, or statistical information.The paper says these interpretations are all relevant to causal understanding, making default choices subtle to justify.Source: §1 · p. 4

2 Causal Models

This section defines causal models through variables and structural equations, which specify how variables respond to interventions. It also sets out assumptions and distinctions that govern how the models are interpreted.

2 Causal Models

  • A causal model consists of a signature listing variables and their possible values, together with modifiable structural equations determining endogenous variables.The forest-fire example distinguishes exogenous factors from modeled variables such as lightning, a dropped match, and fire.Source: §2 · p. 5
  • Structural equations encode intervention effects; setting a variable replaces its equation with the chosen value while leaving the other equations unchanged.For example, the model can specify what happens when an arsonist does not drop a match.Source: §2 · pp. 5–6
  • Once variables are selected, equations can represent objective intervention information, whereas the choice of variables is subjective.The equations characterize intervention results without depending on which values actually occurred.Source: §2 · p. 6
  • The discussion restricts itself to acyclic causal models, in which each context gives a unique solution for the equations.The causal influence can be represented as an acyclic Bayesian network.Source: §2 · pp. 5–6
  • Uncertainty about the causal model and about exogenous-variable values can be represented with probabilities over models and contexts.The resulting framework can express the probability that one event caused another.Source: §2 · p. 6

3 The HP Definition of Actual Cause

The HP account defines actual causes relative to a causal model and context, using counterfactual dependence under contingencies. Its verdicts also depend on whether the model represents variables that distinguish competing causes.

3.1 A language for describing causes

  • Causal formulas combine interventions that set variables to specified values with Boolean claims, evaluated against the model’s unique solution in a context.The forest-fire example illustrates that setting either lightning or the match to absent still leaves the fire, while setting both absent prevents it.Source: §3.1 · p. 7

3.2 A preliminary definition of causality

  • Contingencies let the HP definition classify both lightning and a dropped match as causes of a forest fire despite either one alone failing the naive but-for test.The fire counterfactually depends on each cause when the other is absent.Source: §3.2 · pp. 7–8
  • The preliminary HP test requires the proposed cause and outcome to occur, a contingency under which changing the cause changes the outcome, and minimality of the cause.AC3 removes inessential details from a conjunction; the definition’s verdict is relative to a causal model and context.Source: §3.2 · pp. 8–9
  • With variables for which rock hits the bottle, Suzy’s throw causes the shattering while Billy’s does not when both throw.The model distinguishes simultaneous hits from Suzy’s rock hitting first.Source: §3.2 · p. 11
  • In preemption cases, distinguishing the actual cause requires a modeled variable that differs depending on which event caused the outcome.For competing shots, evidence such as the bullet’s entry side can identify the relevant difference; the variable need not be temporal.Source: §3.2 · pp. 11–12

4 The Choice of Variables

The paper argues that variable selection is not uniquely determined, but can be evaluated with guidelines that reduce arbitrariness. A model’s choices matter because they can make predictions about observations and interventions inaccurate, misleading, or incomplete.

4.1 The Number of Variables

  • Adding variables can change causal verdicts; stability across model extensions and whether added variables alter directed-path topology help assess when a model is sufficient.The authors contrast preemption, where added variables create a new path, with the forest-fire case, where interpolating variables does not change the two-path structure.Source: §4.1 · pp. 12–13

4.2 The Ranges of Variables

  • Variable ranges should capture distinctions needed to represent relationships, such as different orders or strengths of actions and their effects.Three-valued variables can represent conflicting military orders or distinguish a normal push from a solid push when modeling both a door and its alarm.Source: §4.2 · p. 14
  • A variable’s values can presuppose another event, so including Jones’s death as a variable rules out treating his birth as its cause in that model.Both modeled death and survival presuppose that Jones was born.Source: §4.2 · p. 15

4.3 Dependence and Independence

  • Model variables must permit meaningful interventions: logically related values across variables or nonexclusive values within one variable can distort causal judgments.Combining both rocks’ hits into one variable prevents independent interventions needed to distinguish Suzy’s throw as the cause of the bottle shattering.Source: §4.3 · pp. 15–16
  • In the switching example, adding blocked-track variables permits the model to represent a missed arrival and can make throwing the switch a cause under a blockage contingency.The authors note that adding defaults to the definition can later block this causal conclusion.Source: §4.3 · p. 16
  • For the camper’s fire, separate May and June variables avoid cyclic equations and misleading intervention predictions in the alternative representations.In the recommended model, only an intervention on the forest’s May state affects the camper’s plans.Source: §4.3 · p. 17

5 Dealing with normality and typicality

Structural-equation counterfactuals can classify a bodyguard’s antidote as a cause of survival, despite that judgment conflicting with intuition. Adding world normality to the HP account restricts which contingencies count, but leaves the choice of model open to critique.

5 Dealing with normality and typicality

  • The preliminary HP definition treats the bodyguard’s antidote as a cause of survival under a poison contingency, although most people would say it is not.The assassin example is isomorphic to the forest-fire equations, where the analogous causes seem reasonable, showing why structural equations alone do not capture all relevant causal judgments.Source: §5 · p. 18
  • An extended causal model adds a ranking function, allowing AC2(a) contingencies only in worlds no less normal than the actual world.Higher-ranked worlds are less typical; this formalizes the idea that events are more likely to be undone by altering exceptional rather than routine aspects of the causal chain.Source: §5 · p. 19
  • Under the proposed ranking, the poison contingency is less normal than the actual world, so the bodyguard is not a cause of the victim’s survival.The authors report that the definition handles problematic examples, including Example 6.Source: §5 · p. 19
  • In the 100-doctor example, the ranking makes only the assigned doctor a cause of sickness when treatment is omitted, while treatment by the assigned doctor causes recovery.The model assigns different ranks to unassigned, assigned-and-treated, assigned-but-untreated, and other-doctor-treats worlds.Source: §5 · pp. 19–20
  • Normality gives the HP account flexibility, but a modeler can choose an order that makes any proposed cause false, increasing the need to justify the model.The authors say these considerations may not determine one uniquely correct normality theory, but provide resources for rational model critique.Source: §5 · pp. 20–21
  • Normality judgments may draw on statistical frequency, moral views, institutional policies, proper functioning, and legal standards.The cited examples connect these norms to judgments about accidents, miscarriage, pen-taking, mechanical or biological operation, and legal responsibility.Source: §5 · pp. 20–21

6 Conclusion

Causality is relative to a model, so choosing a reasonable model matters. The authors take preliminary steps on variables and defaults, while calling for more work before causal models are used in applications.

6 Conclusion

  • Causality is relative to a model, and the authors’ preliminary work on variables and defaults leaves more to do before causal models are used in applications.They emphasize the need to justify model choices and articulate principles for reasonable causal models.Source: §6 · p. 21
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