Causal Model: The Representation of Causal Structures

• Causal Graph
• Causal Model
• Representation of Probability

For a set of variables \(V\), its causal structure can be represented by a directed acyclic graph (DAG), where each node corresponds to an element of \(V\) while each direct functional relationship among the corresponding variables can be represented by a link in the DAG. A causal structure guides the precise specification of how each variable is influenced by its parents in the DAG. For an instance, \(X \leftarrow W \rightarrow Y\) denotes that \(W\) is a parent, thus also a common cause, of \(X\) and \(Y\). More specifically, for two distinct variables \(V_i\) and \(V_j\), if their functional relationship is

\[V_j = f(V_i, \eta_{ij})\]

for some function \(f\) and noise \(\eta\), then in the DAG representing the causal structure of the set of variables \(V\), there should be an arrow pointing to \(V_j\) from \(V_i\). A detailed introduction to such DAGs for causal structures can be found in [Pearl].

A causal effect, also named as causal estimand, can be expressed with the \(do\)-operator according to [Pearl]. As an example,

\[P(y|do(x))\]

denotes the probability function of \(y\) after imposing the intervention \(x\). Causal structures are crucial to expressing and estimating interested causal estimands. YLearn implements an object, CausalGraph, to support representations for causal structures and related operations of the causal structures. Please see Causal Graph for details.

YLearn concerns the intersection of causal inference and machine learning. Therefore we assume that we have abundant observational data rather than having the access to design randomized experiments. Then Given a DAG for some causal structure, the causal estimands, e.g., the average treatment effects (ATEs), usually can not be directly estimated from the data due to the counterfactuals which can never be observed. Thus it is necessary to convert these causal estimands into other quantities, which can be called as statistical estimands and can be estimated from data, before proceeding to any estimation. The procedure of converting a causal estimand into the corresponding statistical estimand is called identification.

The object for supporting identification and other related operations of causal structures is CausalModel. More details can be found in Causal Model.

In the language of Pearl’s causal inference, it is also necessary to represent the results in the language of probability. For this purpose, YLearn also implements an object Prob which is introduced in Representation of Probability.