For mixed graphs, the ideal corresponding to all trek separation statements A,B,CA,CB (where A,B,CA,CB are disjoint lists of vertices of G) is generated by the r+1 x r+1 minors of the submatrix of the covariance matrix M = (s(i,j)), whose rows are in A, and whose columns are in B, and where r = #CA+#CB.
These ideals are described in more detail by Sullivant, Talaska and Draisma in "Trek Separation for Gaussian Graphical Models" Annals of Statistics 38 no.3 (2010) 1665--1685 and give all determinantal constraints on the covariance matrix of a Gaussian graphical model.
i1 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}})
o1 = MixedGraph{Bigraph => Bigraph{a => set {d}} }
d => set {a}
Digraph => Digraph{b => set {c, d}}
c => set {d}
d => set {}
Graph => Graph{}
o1 : MixedGraph
|
i2 : R = gaussianRing G o2 = R o2 : PolynomialRing |
i3 : T = trekIdeal(R,G)
o3 = ideal (s , s , - s s + s s , s s - s s , -
a,b a,c a,c b,b a,b b,c a,c b,b a,b b,c
------------------------------------------------------------------------
s s + s s , - s s + s s )
a,c b,c a,b c,c a,c b,d a,b c,d
o3 : Ideal of R
|
i4 : ideal gens gb T
o4 = ideal (s , s )
a,c a,b
o4 : Ideal of R
|
For undirected graphs G, the trekIdeal(R,G) is the same as conditionalIndependenceIdeal(R,globalMarkov(G)). For directed graphs G, trekIdeal(R,G) is generally larger than conditionalIndependenceIdeal(R,globalMarkov(G)).
i5 : G = graph{{a,b},{b,c},{c,d},{a,d}}
o5 = Graph{a => set {b, d}}
b => set {a, c}
c => set {b, d}
d => set {a, c}
o5 : Graph
|
i6 : R = gaussianRing G o6 = R o6 : PolynomialRing |
i7 : T = trekIdeal(R,G); o7 : Ideal of R |
i8 : CI = conditionalIndependenceIdeal(R,globalMarkov(G)); o8 : Ideal of R |
i9 : T == CI o9 = true |
i10 : H = digraph{{1,{4}},{2,{4}},{3,{4,5}},{4,{5}}}
o10 = Digraph{1 => set {4} }
2 => set {4}
3 => set {4, 5}
4 => set {5}
5 => set {}
o10 : Digraph
|
i11 : R = gaussianRing H o11 = R o11 : PolynomialRing |
i12 : T = trekIdeal(R,H); o12 : Ideal of R |
i13 : CI = conditionalIndependenceIdeal(R,globalMarkov(H)); o13 : Ideal of R |
i14 : T == CI o14 = false |