List of matrices whose 2x2 minors form the conditional independence ideal of the independence statements on the list S. This method is used in conditionalIndependenceIdeal, it is exported to be able to read independence constraints as minors of matrices instead of their polynomial expansions.
i1 : S = {{{1},{3},{4}}} o1 = {{{1}, {3}, {4}}} o1 : List |
i2 : R = markovRing (4:2) o2 = R o2 : PolynomialRing |
i3 : compactMatrixForm =false; |
i4 : netList markovMatrices (R,S) +--------------------------------------------+ o4 = || p + p p + p || || 1,1,1,1 1,2,1,1 1,1,2,1 1,2,2,1 || || || || p + p p + p || || 2,1,1,1 2,2,1,1 2,1,2,1 2,2,2,1 || +--------------------------------------------+ || p + p p + p || || 1,1,1,2 1,2,1,2 1,1,2,2 1,2,2,2 || || || || p + p p + p || || 2,1,1,2 2,2,1,2 2,1,2,2 2,2,2,2 || +--------------------------------------------+ |
Here is an example where the independence statements are extracted from a graph.
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 : S = localMarkov G o6 = {{{a}, {c}, {d, b}}, {{b}, {d}, {c, a}}} o6 : List |
i7 : R = markovRing (4:2) o7 = R o7 : PolynomialRing |
i8 : markovMatrices (R,S,vertices G) o8 = {| p p |, | p p |, | p p |, | 1,1,1,1 1,1,2,1 | | 1,1,1,2 1,1,2,2 | | 1,2,1,1 1,2,2,1 | | | | | | | | p p | | p p | | p p | | 2,1,1,1 2,1,2,1 | | 2,1,1,2 2,1,2,2 | | 2,2,1,1 2,2,2,1 | ------------------------------------------------------------------------ | p p |, | p p |, | p p |, | 1,2,1,2 1,2,2,2 | | 1,1,1,1 1,1,1,2 | | 1,1,2,1 1,1,2,2 | | | | | | | | p p | | p p | | p p | | 2,2,1,2 2,2,2,2 | | 1,2,1,1 1,2,1,2 | | 1,2,2,1 1,2,2,2 | ------------------------------------------------------------------------ | p p |, | p p |} | 2,1,1,1 2,1,1,2 | | 2,1,2,1 2,1,2,2 | | | | | | p p | | p p | | 2,2,1,1 2,2,1,2 | | 2,2,2,1 2,2,2,2 | o8 : List |
In case the random variables are not numbered 1, 2, …, n, then this method requires an additional input in the form of a list of the random variable names. This list must be in the same order as the implicit order used in the sequence d. The user is encouraged to read the caveat on the method conditionalIndependenceIdeal regarding probability distributions on discrete random variables that have been labeled arbitrarily.