# discreteVanishingIdeal -- vanishing ideal of a discrete graphical model

## Synopsis

• Usage:
discreteVanishingIdeal(R,G)
• Inputs:
• Outputs:
• an ideal, an ideal in $R$

## 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