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discreteVanishingIdeal -- vanishing ideal of a discrete graphical model

Synopsis

Description

This method computes the ideal in $R$ of homogeneous polynomial relations on the joint probabilities of random variables represented by the vertices of $G$.

Here is a small example that compute the vanishing ideal on the joint probabilities of two independent binary random variables. In this case, this ideal equals the ideal obtained using conditionalIndependenceIdeal.

i1 : G = digraph {{1,{}}, {2,{}}}

o1 = Digraph{1 => {}}
             2 => {}

o1 : Digraph
 
i2 : R = markovRing (2,2)

o2 = R

o2 : PolynomialRing
 
i3 : discreteVanishingIdeal (R,G)

o3 = ideal(p   p    - p   p   )
            1,2 2,1    1,1 2,2

o3 : Ideal of R
 
i4 : conditionalIndependenceIdeal(R, localMarkov G)

o4 = ideal(- p   p    + p   p   )
              1,2 2,1    1,1 2,2

o4 : Ideal of R

Here is an example for a graph on four vertices. The random variables a,b,c and d have 2,3,4, and 2 states, respectively.

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 = markovRing (2,3,4,2)

o6 = R

o6 : PolynomialRing
 
i7 : I = discreteVanishingIdeal (R,G);

o7 : Ideal of R

The vanishing ideal is generated by 84 quadrics, which we don't display.

i8 : betti I

            0  1
o8 = total: 1 84
         0: 1  .
         1: . 84

o8 : BettiTally

See also

Ways to use discreteVanishingIdeal:

For the programmer

The object discreteVanishingIdeal is a method function.