computes the structural intervention distance (SID) between two graphs; it evaluates graphs from a causal inference point of view.

1 | ```
structIntervDist(trueGraph, estGraph, output = FALSE, spars = FALSE)
``` |

`trueGraph` |
p x p adjacecncy matrix; it must be a directed acyclic graph (DAG) |

`estGraph` |
again a p x p adjacecncy matrix; it can be either a DAG or a completed partially directed graph (CPDAG) that represents a Markov equivalence class. |

`output` |
boolean (TRUE/FALSE) that indicates whether output shall be shown |

`spars` |
boolean (TRUE/FALSE) that indicates whether the input matrices trueGraph and estGraph are sparse. If matrices are indeed large and sparse this increases the speed of the algorithm. |

The structural intervention distance (SID) considers each pair (i,j) and checks whether the parents of i in estGraph is a valid adjustment set in trueGraph for the causal effect from i to j. If it is, estimating the causal effect from i to j using parent adjustment in estGraph is a correct procedure, independently of the distribution. If it is not, the pair (i,j) contributes a mistake of one to the overall structural intervention distance.

The SID satisfies the properties of a pre-metric; we have SID(G,G) = 0 and 0 <= SID(G1,G2) <= p(p-1), where p is the number of nodes. The SID is not symmetric: in general, we have SID(G1, G2) != SID(G2,G1). Furthermore, the SID can be zero although the two graphs are not the same.

If estGraph is a completed partially directed acyclic graph (CPDAG), that is it describes a Markov equivalence class, the SID outputs a lower and an upper bound of the SID, that correspond to the best and worst DAG in the equivalence class, respectively.

The main function is structIntervDist. It takes two DAGs (adjacency matrices) of the same dimension as input and provides a list with

`sid ` |
the SID |

`sidUpperBound ` |
relevant if estGraph is a CPDAG: it is the highest SID that can be achieved by a graph within the Markov equivalence class ("worst DAG") |

`sidLowerBound ` |
relevant if estGraph is a CPDAG: it is the lowest SID that can be achieved by a graph within the Markov equivalence class ("best DAG") |

`incorrectMat ` |
the matrix of incorrect interventional distributions; its sum equals $sid |

Jonas Peters <jonas.peters@tuebingen.mpg.de>

J. Peters, P. B\"uhlmann: Structural intervention distance (SID) for evaluating causal graphs, Neural Computation 27, pages 771-799, 2015

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | ```
G <- rbind(c(0,1,1,1,1),
c(0,0,1,1,1),
c(0,0,0,0,0),
c(0,0,0,0,0),
c(0,0,0,0,0))
H1 <- rbind(c(0,1,1,1,1),
c(0,0,1,1,1),
c(0,0,0,1,0),
c(0,0,0,0,0),
c(0,0,0,0,0))
H2 <- rbind(c(0,0,1,1,1),
c(1,0,1,1,1),
c(0,0,0,0,0),
c(0,0,0,0,0),
c(0,0,0,0,0))
cat("true DAG G:\n")
show(G)
cat("\n")
cat("==============","\n")
cat("\n")
cat("estimated DAG H1:\n")
show(H1)
sid <- structIntervDist(G,H1)
shd <- hammingDist(G,H1)
cat("SHD between G and H1: ",shd,"\n")
cat("SID between G and H1: ",sid$sid,"\n")
#cat("The matrix of incorrect interventional distributions is: ","\n")
#show(sid$incorrectMat)
cat("\n")
cat("==============","\n")
cat("\n")
cat("estimated DAG H2:\n")
show(H2)
sid <- structIntervDist(G,H2)
shd <- hammingDist(G,H2)
cat("SHD between G and H2: ",shd,"\n")
cat("SID between G and H2: ",sid$sid,"\n")
cat("The matrix of incorrect interventional distributions is: ","\n")
show(sid$incorrectMat)
readline("The SID can also be applied to CPDAGs. Please press enter...")
cat("\n")
cat("==============","\n")
cat("\n")
cat("estimated CPDAG H1c:\n")
H1c <- rbind(c(0,1,1,1,1),c(1,0,1,1,1),c(1,1,0,1,0),c(1,1,1,0,0),c(1,1,0,0,0))
show(H1c)
sid <- structIntervDist(G,H1c)
cat("SID between G and H1: \n")
cat("lower bound: ",sid$sidLowerBound," upper bound: ", sid$sidUpperBound, "\n")
cat("\n")
``` |

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