cfid (version 0.1.8)

identifiable: Identify a Counterfactual Query

Description

Determine the identifiability of a (conditional) counterfactual conjunction.

Usage

identifiable(
  g,
  gamma,
  delta = NULL,
  data = c("interventions", "observations", "both")
)

Value

An object of class query which is a list containing one or more of the following:

• id
  A logical value that is TRUE if the query is identifiable and FALSE otherwise from the available data in g. Note that in cases where gamma itself is inconsistent, the query will be identifiable, but with probability 0.

• formula
  An object of class functional giving the identifying functional of the query in LaTeX syntax via format or print, if identifiable. This expression is given in terms of the available data. Variables bound by summation are distinguished by a superscript asterisk. For tautological statements, the resulting probability is 1, and for inconsistent statements, the resulting probability is 0. For formatting options, see format.functional() and format.probability().

• undefined
  A logical value that is TRUE if a conditional conjunction \(p(\gamma|\delta)\) is undefined, for example when \(p(\delta) = 0\), and FALSE otherwise.

• gamma
  The original counterfactual conjunction..

• delta
  The original conditioning counterfactual conjunction.

• data
  The original data.

Arguments

g

A dag object describing the causal graph (to obtain a dag from another format, see import_graph().

gamma

An R object that can be coerced into a counterfactual_conjunction object that represents the counterfactual causal query.

delta

An R object that can be coerced into a counterfactual_conjunction object that represents the conditioning conjunction (optional).

data

A character string that accepts one of the following: "interventions" (the default), "observations"or "both". This argument defines the target level of identification. If "interventions" is used, the identification is attempted down to the intervention level. If "observations" is used, identification is attempted down to the observational level. If "both" is used, identification is carried out for each term to the lowest level where the term is still identifiable.

Details

To identify a non-conditional conjunction \(P(\gamma)\), the argument delta should be NULL.

To identify a conditional conjunction \(P(\gamma|\delta)\), both gamma and delta should be specified.

First, a parallel worlds graph is constructed based on the query. In a parallel worlds graph, for each \(do\)-action that appears in \(\gamma\) (and \(\delta\)) a copy of the original graph is created with the new observational variables attaining their post-interventional values but sharing the latent variables. This graph is known as a parallel worlds graph. From the parallel worlds graph, a counterfactual graph is derived such that each variable is unique, which might not be the case in a parallel worlds graph.

Finally, the ID* (or IDC*) algorithm is applied to determine identifiability of the query. Similar to the ID and IDC algorithms for causal effects, these algorithms exploit the so called c-component factorization to split the query into smaller subproblems, which are then solved recursively. If argument data is "observations" or "both", identification of interventional probabilities in the resulting functional is further attempted in terms of the joint probability distribution by using the ID and IDC algorithms (see causal_effect).

See Also

dag(), counterfactual_variable(), probability(), functional()

Examples

# Examples that appears in Shpitser and Pearl (2008)
g1 <- dag("X -> W -> Y <- Z <- D X <-> Y")
g2 <- dag("X -> W -> Y <- Z <- D X <-> Y X -> Y")
v1 <- cf("Y", 0, c(X = 0))
v2 <- cf("X", 1)
v3 <- cf("Z", 0, c(D = 0))
v4 <- cf("D", 0)
c1 <- conj(v1)
c2 <- conj(v2, v3, v4)
c3 <- conj(v1, v2, v3, v4)

# Identifiable conditional conjunction
identifiable(g1, c1, c2)

# Identifiable conjunction
identifiable(g1, c3)

# Non-identifiable conjunction
identifiable(g2, c3)