Categories → Causal Inference , Statistics , Study Design
The do-calculus is a set of mathematical rules for reasoning about causality in graphical models.
For a quick and easy introduction to the do-Calculus, check out this video by Brady Neal. It's great.
The purpose of the do-calculus is to provide a systematic way to answer causal questions. In causal inference, we are interested in understanding the causal relationships between variables. However, the relationships between variables can be complex and often involve confounding variables that make it difficult to establish causal relationships through observation alone.
The do-calculus allows us to manipulate the variables in a causal graph to simulate the effects of interventions, or actions, on the system. By performing interventions, we can isolate the causal relationships between variables and eliminate the confounding effects of other variables.
The do-calculus consists of three key rules: the intervention rule, the composition rule, and the reduction rule. These rules allow us to compute the effects of interventions on the probability distribution of the variables in the graph.
The intervention rule tells us how to compute the effect of intervening on a variable, while the composition rule allows us to combine multiple interventions. The reduction rule allows us to simplify the graph by eliminating variables that are not relevant to the causal question we are interested in.
In the context of causal inference, the do-calculus is considered both sufficient and complete. These important claims were only recently confirmed (see paper).
Sufficiency means that the do-calculus provides a set of rules that are capable of capturing all possible causal relationships between variables in a causal graph. In other words, the do-calculus is powerful enough to represent any causal relationship that could exist in a system, as long as it can be represented in a graphical model.
Completeness means that the do-calculus provides a set of rules that are capable of answering any causal question that can be formulated in terms of the graph's variables. In other words, the do-calculus is a complete framework for causal inference, meaning that there is no causal question that cannot be answered using its rules.
The sufficiency and completeness of the do-calculus are important because they give us confidence that we can use this framework to reason about causality in any system and not miss out on possible solutions. With the do-calculus, we can specify a causal model using a graphical representation, and then use the rules of the do-calculus to reason about the causal relationships between variables and simulate the effects of interventions.
It is worth noting that the do-calculus assumes that the causal graph is acyclic, meaning that there are no feedback loops in the system. This assumption allows us to reason about the effects of interventions in a straightforward manner. However, in some real-world systems, feedback loops are present, and the do-calculus may not be applicable in these cases. Nonetheless, in acyclic causal models, the do-calculus is a powerful and complete framework for causal inference.
It may be possible to break these loops by restricting your modelling of the system to a before / after setting.
Overall, the do-calculus is a powerful tool for causal inference because it allows us to reason about the causal relationships between variables and simulate the effects of interventions, even in complex systems with many variables and confounding factors.