# Identify a Counterfactual Query Determine the identifiability of a (conditional) counterfactual conjunction. ```r identifiable( g, gamma, delta = NULL, data = c("interventions", "observations", "both") ) ``` ## Arguments - `g`: A `dag` object describing the causal graph (to obtain a `dag` from another format, see `import_graph()`. - `gamma`: An object that can be coerced into a `counterfactual_conjunction` object that represents the counterfactual causal query. - `delta`: An 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. ## Returns 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. ## 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 ). ## Examples ```r # 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) ``` ## See Also `dag()`, `counterfactual_variable()`, `probability()`, `functional()`

identifiable function