gaussianVanishingIdeal -- vanishing ideal of a Gaussian graphical model

Synopsis

Usage:

gaussianVanishingIdeal(R)
Inputs:
- R, a ring, created with gaussianRing using a graphs of classes Graph, Digraph or MixedGraph without undirected edges as input.
Optional inputs:
- OldVersion => ..., default value false, optional argument in gaussianVanishingIdeal to use old method for gaussianRings coming from directed graphs
Outputs:
- an ideal, ideal in R

Description

gaussianVanishingIdeal computes the ideal in $R$ of homogeneous polynomial relations on the variance-covariance parameters of a graphical model on $G$ as explained in Chapter 3.3 of ``Lectures on Algebraic Statistics'' by Drton, Sturmfels, and Sullivant.

i1 : G = graph({{a,b},{b,c},{c,d},{a,d}})

o1 = Graph{a => {b, d}}
           b => {a, c}
           c => {b, d}
           d => {a, c}

o1 : Graph

i2 : R = gaussianRing G

o2 = R

o2 : PolynomialRing

i3 : J = gaussianVanishingIdeal(R);

o3 : Ideal of R

i4 : ideal mingens J / print;
                    2
s   s   s    - s   s    - s   s   s    + s   s   s    + s   s   s    - s   s   s
 a,d b,c b,d    a,c b,d    a,d b,b c,d    a,b b,d c,d    a,c b,b d,d    a,b b,c d,d
                2
s   s   s    - s   s    - s   s   s    + s   s   s    + s   s   s    - s   s   s
 a,c a,d b,c    a,c b,d    a,b a,d c,c    a,a b,d c,c    a,b a,c c,d    a,a b,c c,d

This method works for graphs, digraphs and mixed graphs without undirected edges.

i5 : G = digraph {{a,{b,c}}, {b,{c,d}}, {c,{}}, {d,{}}}

o5 = Digraph{a => {c, b}}
             b => {c, d}
             c => {}
             d => {}

o5 : Digraph

i6 : R = gaussianRing G

o6 = R

o6 : PolynomialRing

i7 : gaussianVanishingIdeal(R)

o7 = ideal (s   s    - s   s   , s   s    - s   s   , s   s    - s   s   )
             b,c b,d    b,b c,d   a,d b,c    a,b c,d   a,d b,b    a,b b,d

o7 : Ideal of R

i8 : H = mixedGraph(digraph {{a,{c}},{b,{c}}, {c,{d}}},bigraph {{c,d}})

o8 = MixedGraph{Bigraph => Bigraph{c => {d}}}
                                   d => {c}
                Digraph => Digraph{a => {c}}
                                   b => {c}
                                   c => {d}
                                   d => {}
                Graph => Graph{}

o8 : MixedGraph

i9 : S = gaussianRing H

o9 = R

o9 : PolynomialRing

i10 : gaussianVanishingIdeal(S)

o10 = ideal (s   s    - s   s   , s   s    - s   s   , s   s    - s   s   )
              b,c b,d    b,b c,d   a,d b,c    a,b c,d   a,d b,b    a,b b,d

o10 : Ideal of R

Caveat

This method currently works on really small examples because it computes the vanishing ideal as an elimination ideal.

Ways to use gaussianVanishingIdeal:

gaussianVanishingIdeal(Ring)

For the programmer

The object gaussianVanishingIdeal is a method function with options.