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# trekIdeal -- trek separation ideal of a mixed graph

- Usage:
    `I = trekIdeal(R,G)`
- Inputs:
  - R, a ring, which should be a gaussianRing
  - G, a graph, a graph, or a digraph with no cycles, or an instance of the type MixedGraph with directed and bidirected edges
- Outputs:
  - I, an ideal, the ideal of determinantal trek separation statements implied by the graph $G$.

## Description

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.

This function is not yet implemented for mixed graphs with undirected edges.

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 => {d}}   }
                                       d => {a}
                    Digraph => Digraph{b => {c, d}}
                                       c => {d}
                                       d => {}
                    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 => {b, d}}
               b => {a, c}
               c => {b, d}
               d => {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 => {4}   }
                  2 => {4}
                  3 => {4, 5}
                  4 => {5}
                  5 => {}

    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

## Caveat

trekSeparation is currently only implemented with mixedGraphs that have directed and bidirected edges.

## See also

- trekSeparation -- trek separation statements of a mixed graph

## Ways to use trekIdeal:

- `trekIdeal(Ring,Digraph)`
- `trekIdeal(Ring,Graph)`
- `trekIdeal(Ring,MixedGraph)`

## For the programmer

The object trekIdeal is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GraphicalModels.m2:2643:0.