pc.cons.intern: Utility for conservative and majority rule in PC and FCI

Description

The pc.cons.intern() function is used in pc and fci, notably when conservative = TRUE (conservative orientation of v-structures) or maj.rule = TRUE (majority rule orientation of v-structures).

Usage

pc.cons.intern(sk, suffStat, indepTest, alpha, version.unf = c(NA, NA), maj.rule = FALSE, verbose = FALSE)

Arguments

A skeleton object as returned from skeleton().

suffStat

Sufficient statistic: List containing all necessary elements for the conditional independence decisions in the function indepTest.

indepTest

Pre-defined function for testing conditional independence. The function is internally called as indepTest(x,y,S,suffStat), and tests conditional independence of x and y given S. Here, x and y are variables, and S is a (possibly empty) vector of variables (all variables are denoted by their column numbers in the adjacency matrix). suffStat is a list containing all relevant elements for the conditional independence decisions. The return value of indepTest is the p-value of the test for conditional independence.

alpha

Significance level for the individual conditional independence tests.

version.unf

Vector of length two. If version.unf[2]==1, the intitial separating set found by the PC/FCI algorithm is added to the set of separating sets; if version.unf[2]==2, it is not added. In the latter case, if the set of separating sets is empty, the triple is marked as unambiguous if version.unf[1]==1, and as ambiguous if version.unf[1]==2.

maj.rule

Logical indicatin if the triples are checked for ambiguity using the majority rule idea, which is less strict than the standard conservative method.

verbose

Logical asking for detailed output.

Value

Details

For any unshielded triple A-B-C, consider all subsets of the neighbors of A and of the neighbors of C, and record all such sets D for which A and C are conditionally independent given D. We call such sets “separating sets”.

If version.unf[2]==1, the initial separating set found in the PC/FCI algorithm is added to this set of separating sets. If version.unf[2]==2, the initial separating set is not added (as in Tetrad).

In the latter case, if the set of separating sets is empty, then the triple is marked as ‘ambiguous’ if version.unf[1]==2, for example in pc, or as ‘unambiguous’ if version.unf[1]==1, for example in fci. Otherwise, there is at least one separating set. If maj.rule=FALSE, the conservative PC algorithm is used (Ramsey et al., 2006): If B is in some but not all separating sets, the triple is marked as ambiguous. Otherwise it is treated as in the standard PC algorithm. If maj.rule=TRUE, the majority rule is applied (Colombo and Maathuis, 2014): The triple is marked as ‘ambiguous’ if B is in exactly 50 percent of the separating sets. If it is in less than 50 percent it is marked as a v-structure, and if it is in more than 50 percent it is marked as a non v-structure.

Note: This function modifies the separating sets for unambiguous triples in the skeleton object (adding or removing B) to ensure that the usual orientations rules later on lead to the correct v-structures/non v-structures.

References

D. Colombo and M.H. Maathuis (2014).Order-independent constraint-based causal structure learning. Journal of Machine Learning Research 15 3741-3782.

J. Ramsey, J. Zhang and P. Spirtes (2006). Adjacency-faithfulness and conservative causal inference. In Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence, Arlington, VA. AUAI Press.