9.4 Interventions and do-calculus

Interventions and do-calculus are crucial tools in causal inference. They help us distinguish between causal relationships and mere associations by actively changing variables and observing outcomes. These concepts form the foundation for designing experiments and analyzing data to establish cause-and-effect relationships.

Do-calculus, developed by Judea Pearl, provides a formal framework for reasoning about interventions. It extends standard probability theory by incorporating interventional distributions and causal assumptions. This allows researchers to identify causal effects from observational data and solve complex identifiability problems in causal inference.

Defining interventions

• Interventions are a fundamental concept in causal inference that involve actively changing the value of a variable to observe its effect on an outcome
• Understanding interventions is crucial for distinguishing between causal relationships and mere associations or correlations
• Interventions form the basis for designing and analyzing experiments to establish causal effects

Interventional distributions

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• Interventional distributions describe the probability distribution of variables when an intervention is performed on a specific variable
• Unlike observational distributions, interventional distributions are not influenced by the natural relationships between variables
• Interventional distributions allow us to reason about the effects of hypothetical interventions and estimate causal effects

Causal vs observational

• Causal relationships imply that changing one variable directly affects another variable, while observational relationships only indicate an association without necessarily implying causation
• Observational data alone is insufficient to establish causal relationships due to potential confounding factors and selection bias
• Interventions are necessary to distinguish between causal and observational relationships by actively manipulating variables and observing their effects

Ideal randomized experiments

• Ideal randomized experiments are the gold standard for establishing causal relationships by randomly assigning subjects to different treatment groups
• Randomization ensures that confounding factors are balanced across treatment groups, allowing for unbiased estimation of causal effects
• Examples of ideal randomized experiments include clinical trials (drug testing) and A/B testing (website design)

Representing interventions

• To reason about interventions and their effects, we need a formal way to represent them mathematically and graphically
• Representing interventions allows us to express causal assumptions, identify causal effects, and communicate our understanding of the causal structure

Intervention notation

• The do-operator, denoted as $do(X=x)$, represents an intervention that sets the variable $X$ to a specific value $x$
• The do-operator distinguishes between observational and interventional distributions, with $P(Y|do(X=x))$ representing the distribution of $Y$ when $X$ is set to $x$ through an intervention
• Intervention notation allows us to express causal queries and estimate causal effects using mathematical expressions

Graphical representation

• Causal graphs, such as directed acyclic graphs (DAGs), visually represent the causal relationships between variables
• In a causal graph, nodes represent variables, and directed edges represent direct causal effects
• Interventions can be represented graphically by removing incoming edges to the intervened variable, indicating that its value is determined by the intervention rather than its natural causes

Manipulated graphs

• Manipulated graphs, also known as mutilated graphs, are causal graphs that have been modified to represent the effects of interventions
• When an intervention is performed on a variable, the manipulated graph is obtained by removing all incoming edges to the intervened variable
• Manipulated graphs allow us to reason about the effects of interventions and identify causal effects using graphical criteria (back-door criterion, front-door criterion)

Identifiability under interventions

• Identifiability refers to the ability to estimate causal effects from observational data under certain assumptions about the causal structure
• Identifying causal effects is crucial for making informed decisions and predicting the outcomes of interventions
• Several graphical criteria and methods have been developed to assess the identifiability of causal effects under interventions

Back-door criterion

• The back-door criterion is a graphical condition that allows for the identification of causal effects by adjusting for a sufficient set of variables that block all back-door paths between the treatment and outcome
• Back-door paths are non-causal paths that connect the treatment and outcome through a common cause (confounder)
• Adjusting for a set of variables satisfying the back-door criterion removes confounding bias and enables the estimation of causal effects (smoking -> cancer, adjusting for age)

Front-door criterion

• The front-door criterion is a graphical condition that allows for the identification of causal effects when there are unmeasured confounders between the treatment and outcome
• The front-door criterion requires the existence of a mediator variable that is influenced by the treatment and affects the outcome, while being independent of the unmeasured confounders
• By adjusting for the mediator variable, the front-door criterion enables the estimation of causal effects in the presence of unmeasured confounding (education -> income, adjusting for skill level)

Instrumental variables

• Instrumental variables (IVs) are variables that affect the treatment but not the outcome directly, and are independent of unmeasured confounders
• IVs can be used to estimate causal effects when there is unmeasured confounding between the treatment and outcome
• By leveraging the variation in the treatment induced by the IV, causal effects can be estimated under certain assumptions (using proximity to colleges as an IV for education -> income)

Intro to do-calculus

• Do-calculus is a formal framework for reasoning about interventions and identifying causal effects from observational and interventional distributions
• Developed by Judea Pearl, do-calculus provides a set of rules for manipulating probability distributions under interventions
• Understanding do-calculus is essential for advanced causal inference and solving complex identifiability problems

Formal definition

• Do-calculus consists of three rules that allow for the manipulation of interventional distributions and the identification of causal effects
• The rules are based on the graphical structure of the causal model and the independence relationships encoded in the graph
• The formal definition of do-calculus provides a rigorous foundation for causal reasoning and inference

Differences from standard probability

• Do-calculus extends standard probability theory by incorporating interventional distributions and causal assumptions
• In do-calculus, probability distributions are manipulated under interventions, which differs from the manipulation of conditional probabilities in standard probability theory
• Do-calculus allows for the expression of counterfactual queries and the identification of causal effects, which is not possible with standard probability alone

Three rules of do-calculus

• Rule 1 (Insertion/Deletion of Observations): $P(y|do(x), z, w) = P(y|do(x), w)$ if $(Y \perp Z | X, W)_{G_{\overline{X}}}$
• Rule 2 (Action/Observation Exchange): $P(y|do(x), do(z), w) = P(y|do(x), z, w)$ if $(Y \perp Z | X, W)_{G_{\overline{X}\underline{Z}}}$
• Rule 3 (Insertion/Deletion of Actions): $P(y|do(x), do(z), w) = P(y|do(x), w)$ if $(Y \perp Z | X, W)_{G_{\overline{X}, \overline{Z(W)}}}$
• These rules allow for the manipulation of interventional distributions based on the graphical structure and independence relationships

Applying do-calculus

• Do-calculus provides a powerful tool for identifying causal effects and solving complex identifiability problems
• By applying the rules of do-calculus, researchers can determine whether causal effects can be estimated from observational data and derive the necessary formulas for estimation
• Applying do-calculus requires a deep understanding of the causal structure and the ability to manipulate probability distributions under interventions

Identifying causal effects

• Do-calculus can be used to identify causal effects by transforming interventional distributions into observational distributions that can be estimated from data
• By applying the rules of do-calculus, researchers can derive the appropriate adjustment sets or identification formulas for estimating causal effects
• Examples of identifying causal effects include deriving the back-door adjustment formula and the front-door adjustment formula

Proving non-identifiability

• In some cases, causal effects may not be identifiable from observational data due to the presence of unmeasured confounders or other limitations in the causal structure
• Do-calculus can be used to prove the non-identifiability of causal effects by showing that no sequence of do-calculus rules can transform the interventional distribution into an observational distribution
• Proving non-identifiability is important for understanding the limitations of causal inference and the need for additional data or assumptions

Worked examples

• Applying do-calculus to real-world problems involves carefully analyzing the causal structure, identifying the relevant variables and independence relationships, and applying the rules of do-calculus
• Worked examples help to illustrate the process of applying do-calculus and demonstrate its effectiveness in identifying causal effects
• Examples of worked examples include identifying the causal effect of smoking on lung cancer, estimating the effect of education on income, and proving the non-identifiability of the causal effect of race on job hiring

Interventions with SCMs

• Structural Causal Models (SCMs) provide a formal framework for representing and reasoning about interventions in causal systems
• SCMs combine graphical representations of causal relationships with mathematical equations that describe the functional relationships between variables
• Interventions can be naturally incorporated into SCMs, allowing for the analysis of causal effects and counterfactual reasoning

Modeling interventions

• In SCMs, interventions are modeled by replacing the structural equations of the intervened variables with the intervention values
• Interventions can be deterministic (setting a variable to a specific value) or stochastic (drawing values from a probability distribution)
• Modeling interventions in SCMs allows for the computation of interventional distributions and the estimation of causal effects

Truncated factorization

• Truncated factorization is a key concept in SCMs that describes the distribution of variables under interventions
• When an intervention is performed on a variable, the truncated factorization is obtained by removing the structural equation of the intervened variable and using the intervention value instead
• Truncated factorization allows for the computation of interventional distributions and the identification of causal effects in SCMs

Causal effect estimation

• SCMs provide a principled way to estimate causal effects by comparing the outcomes under different interventions
• Causal effects can be estimated by computing the expected difference in outcomes between two interventions, such as the average treatment effect (ATE) or the effect of treatment on the treated (ETT)
• SCMs enable the estimation of causal effects in the presence of confounding, mediation, and other complex causal structures

Practical considerations

• While interventions and do-calculus provide a powerful framework for causal inference, there are several practical considerations to keep in mind when applying these methods in real-world settings
• Researchers must carefully consider the feasibility, ethics, and limitations of interventions and causal inference to ensure the validity and applicability of their findings

Feasibility of interventions

• Not all interventions are feasible or practical to implement in real-world settings due to resource constraints, logistical challenges, or ethical considerations
• Researchers must carefully consider the feasibility of interventions and seek alternative approaches (natural experiments, quasi-experiments) when direct interventions are not possible
• Feasibility considerations include cost, time, sample size, and potential unintended consequences of interventions

Ethical implications

• Interventions can have significant ethical implications, particularly when they involve human subjects or sensitive topics
• Researchers must adhere to ethical guidelines and obtain informed consent from participants when conducting interventional studies
• Ethical considerations include minimizing harm, ensuring fairness and equity, respecting autonomy, and protecting vulnerable populations

Limitations of do-calculus

• While do-calculus is a powerful tool for causal inference, it has some limitations that researchers must be aware of
• Do-calculus relies on the assumptions of the underlying causal model, such as the causal Markov condition and faithfulness, which may not always hold in practice
• The identifiability of causal effects depends on the causal structure and the available data, and some causal effects may not be identifiable even with do-calculus
• Researchers must carefully consider the limitations of do-calculus and use it in conjunction with other methods (sensitivity analysis, bounds) to assess the robustness of their findings