checkTriple function

Check Consistency of Conditional Independence for a Triple of Nodes

For each subset of nbrsA and nbrsC where a and c are conditionally independent, it is checked if b is in the conditioning set.

checkTriple(a, b, c, nbrsA, nbrsC,
            sepsetA, sepsetC,
            suffStat, indepTest, alpha, version.unf = c(NA, NA),
            maj.rule = FALSE, verbose = FALSE)

Arguments

• a, b, c: (integer) positions in adjacency matrix for nodes $a$, $b$, and $c$, respectively.

• nbrsA, nbrsC: (integer) position in adjacency matrix for neighbors of $a$ and $c$, respectively.

• sepsetA: vector containing $Sepset(a,c)$.

• sepsetC: vector containing $Sepset(c,a)$.

• suffStat: a list of sufficient statistics for independent tests; see, e.g., pc.

• indepTest: a function for the independence test, see, e.g., pc.

• alpha: significance level of test.

• version.unf: (integer) vector of length two:

  • version.unf[1]:: 1 - check for all separating subsets of nbrsA and nbrsC if b is in that set,

     2 - it also checks if there at all exists any sepset which is a subset of the neighbours (there might be none, although `b`
     
     is in the sepset, which indicates an ambiguous situation);
    

  • version.unf[2]:: 1 - do not consider the initial sepsets sepsetA and sepsetC (same as Tetrad),

     2 - consider if `b` is in `sepsetA` or `sepsetC`. 
    

• maj.rule: logical indicating that the following majority rule is applied: if b is in less than 50% of the checked sepsets, we say that b is in no sepset. If b is in more than 50% of the checked sepsets, we say that

  b is in all sepsets. If b is in exactly 50% of the checked sepsets, the triple is considered ambiguous .

• verbose: Logical asking for detailed output of intermediate steps.

Details

This function is used in the conservative versions of structure learning algorithms.

Returns

• decision: Decision on possibly ambiguous triple, an integer code,

  • 1: b is in NO sepset (make v-structure);
  • 2: b is in ALL sepsets (make no v-structure);
  • 3: b is in SOME but not all sepsets (ambiguous triple)

• vers: Version (1 or 2) of the ambiguous triple (1=normal ambiguous triple that is b is in some sepsets; 2=triple coming from version.unf[1]==2, that is, a and c are indep given the initial sepset but there doesn't exist a subset of the neighbours that d-separates them.)

• sepsetA: Updated version of sepsetA

• sepsetC: Updated version of sepsetC

References

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

15 3741-3782.

Author(s)

Markus Kalisch (kalisch@stat.math.ethz.ch ) and Diego Colombo.

Examples

##################################################
## Using Gaussian Data
##################################################
## Load predefined data
data(gmG)
n <- nrow    (gmG8$x)
V <- colnames(gmG8$x)

## define independence test (partial correlations), and test level
indepTest <- gaussCItest
alpha <- 0.01
## define sufficient statistics
suffStat <- list(C = cor(gmG8$x), n = n)

## estimate CPDAG
pc.fit <- pc(suffStat, indepTest, alpha=alpha, labels = V, verbose = TRUE)

if (require(Rgraphviz)) {
  ## show estimated CPDAG
  par(mfrow=c(1,2))
  plot(pc.fit, main = "Estimated CPDAG")
  plot(gmG8$g, main = "True DAG")
}

a <- 6
b <- 1
c <- 8
checkTriple(a, b, c,
            nbrsA = c(1,5,7),
            nbrsC = c(1,5),
            sepsetA = pc.fit@sepset[[a]][[c]],
            sepsetC = pc.fit@sepset[[c]][[a]],
            suffStat=suffStat, indepTest=indepTest, alpha=alpha,
            version.unf = c(2,2),
            verbose = TRUE) -> ct
str(ct)
## List of 4
## $ decision: int 2
## $ version : int 1
## $ SepsetA : int [1:2] 1 5
## $ SepsetC : int 1

checkTriple(a, b, c,
            nbrsA = c(1,5,7),
            nbrsC = c(1,5),
            sepsetA = pc.fit@sepset[[a]][[c]],
            sepsetC = pc.fit@sepset[[c]][[a]],
            version.unf = c(1,1),
            suffStat=suffStat, indepTest=indepTest, alpha=alpha) -> c2
stopifnot(identical(ct, c2)) ## in this case,  'version.unf' had no effect