add minimal bd function

src/CausalInference.jl

@@ -4,7 +4,7 @@ using Graphs.SimpleGraphs
 using Combinatorics
 using Base.Iterators
 
-export ancestors, descendants, alt_test_dsep, test_covariate_adjustment, alt_test_backdoor, find_dsep, find_min_dsep, find_covariate_adjustment, find_backdoor_adjustment, find_min_covariate_adjustment, find_frontdoor_adjustment, list_dseps, list_covariate_adjustment, list_backdoor_adjustment, list_frontdoor_adjustment
+export ancestors, descendants, alt_test_dsep, test_covariate_adjustment, alt_test_backdoor, find_dsep, find_min_dsep, find_covariate_adjustment, find_backdoor_adjustment, find_min_covariate_adjustment, find_min_backdoor_adjustment, find_frontdoor_adjustment, list_dseps, list_covariate_adjustment, list_backdoor_adjustment, list_frontdoor_adjustment
 export dsep, skeleton, gausscitest, dseporacle, partialcor
 export unshielded, pcalg, vskel
 export cpdag, alt_cpdag

src/gensearch.jl

@@ -175,6 +175,8 @@ Follows the algorithmic approach proposed in https://arxiv.org/abs/1803.00116.
 """
 function find_backdoor_adjustment(g, X, Y, I = Set{eltype(g)}(), R = setdiff(Set(vertices(g)), X, Y))
     bdZ = setdiff(find_covariate_adjustment(g, X, Y, I, R), descendants(g, X, no_incoming(X)))
+    # we generalize bd to sets by testing (X indep Y given Z) in G with outgoing edges from X removed
+    # instead of checking the bd criterion for pairs of vertices (x in X and y in Y) as originally proposed by Pearl which is more restrictive
     if issubset(I, bdZ) && alt_test_dsep(g, X, Y, bdZ, no_outgoing(X))
         return bdZ
     else
@@ -194,6 +196,16 @@ function find_min_covariate_adjustment(g, X, Y, I = Set{eltype(g)}(), R = setdif
     return find_min_dsep(g, X, Y, I, setdiff(R, descendants(g, PCPXY)), (pe, ne, v, w) -> v in X && w in PCPXY && ne == RIGHT)
 end
 
+"""
+    find_min_backdoor_adjustment(g, X, Y, I = Set{eltype(g)}(), R = setdiff(Set(vertices(g)), X, Y))
+
+Find an inclusion minimal backdoor adjustment set `Z` with `I subseteq Z subseteq R` for sets of vertices `X` and `Y` in `g`, else return `false`. 
+
+"""
+function find_min_backdoor_adjustment(g, X, Y, I = Set{eltype(g)}(), R = setdiff(Set(vertices(g)), X, Y))
+    return find_min_dsep(g, X, Y, I, setdiff(R, descendants(g, X)), (pe, ne, v, w) -> v in X && ne == RIGHT)
+end
+
 """
     find_frontdoor_adjustment(g, X, Y, I = Set{eltype(g)}(), R = setdiff(Set(vertices(g)), X, Y))
 

test/gensearch.jl

@@ -8,7 +8,7 @@ Y = Set(4)
 
 # for some of the tests other results would not be wrong per se 
 # e.g. for the finding methods, other valid sets could be returned
-# however, the algorithms should be deterministic, so I think the
+# however, the algorithms should be deterministic, so we think the
 # tests are okay for now
 
 @testset "gensearch in1" begin
@@ -31,6 +31,7 @@ Y = Set(4)
     @test find_covariate_adjustment(g, X, Y) == Set([5,6])
     @test find_backdoor_adjustment(g, X, Y) == Set([5,6])
     @test find_min_covariate_adjustment(g, X, Y) ==  Set(5)
+    @test find_min_backdoor_adjustment(g, X, Y) == Set(5)
     @test find_frontdoor_adjustment(g, X, Y) == Set([2,3,7])
     @test Set(list_dseps(g, X, Y)) == Set([Set([3,6]), Set([2,6]), Set([2,3,6]), Set([3,6,7]), Set([2,6,7]), Set([2,3,6,7]), Set([3,5]), Set([2,5]), Set([2,3,5]), Set([3,5,7]), Set([2,5,7]), Set([2,3,5,7]), Set([3,5,6]), Set([2,5,6]), Set([2,3,5,6]), Set([3,5,6,7]), Set([2,5,6,7]), Set([2,3,5,6,7])])
     @test Set(list_covariate_adjustment(g, X, Y)) == Set([Set([5,7]), Set([5]), Set([6]), Set([6,7]), Set([5,6]), Set([5,6,7])])