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scoreEquations(...,Saturate=>...) -- whether to saturate

Description

scoreEquations(...,Saturate=>...) is set to true by default. If b is false, saturation is not performed.

Avoiding saturation is only intended for big computations when saturation cannot be computed or the computational time is very high. When Saturate is set to false, scoreEquations might not output the ideal generated by score equations because of the existence of vanishing denominators.

For graphs with only undirected edges, the ideal of score equations is the saturation of the outputted ideal by the determinant of the concentration matrix. In the general case, the ideal of score equations consist of the saturation of the outputted ideal by the product of denominators of the Jacobian matrix.

For example, in the following case the degree of the ideal prior to saturation already gives the right ML-degree:

i1 : G = mixedGraph(digraph {{1,2},{1,3},{2,3},{3,4}},bigraph {{3,4}});
 
i2 : R=gaussianRing(G);
 
i3 : U = matrix{{6, 10, 1/3, 1}, {3/5, 3, 1/2, 1}, {4/5, 3/2, 9/8, 3/10}, {10/7, 2/3,1, 8/3}};

              4       4
o3 : Matrix QQ  <-- QQ
 
i4 : JnoSat=scoreEquations(R,U,Saturate=>false);

o4 : Ideal of QQ[l   ..l   , l   , l   , p   , p   , p   , p   , p   ]
                  1,2   1,3   2,3   3,4   1,1   2,2   3,3   4,4   3,4
 
i5 : dim JnoSat

o5 = 3
 
i6 : degree JnoSat

o6 = 1
 
i7 : J=scoreEquations(R,U)

o7 = ideal (192199680p    - 99333449, 267221621760p    - 849243924773,
                      3,4                          4,4                
     ------------------------------------------------------------------------
     1353974896462794079472640p    - 142165262245288892244817, 6898968p    -
                               3,3                                     2,2  
     ------------------------------------------------------------------------
     11533057, 19600p    - 95819, 20855l    + 90447,
                     1,1                3,4         
     ------------------------------------------------------------------------
     146915678869660815915l    - 4228634793402814499,
                           2,3                       
     ------------------------------------------------------------------------
     58766271547864326366l    + 4167005135395196717, 574914l    - 896035)
                          1,3                               1,2

o7 : Ideal of QQ[l   ..l   , l   , l   , p   , p   , p   , p   , p   ]
                  1,2   1,3   2,3   3,4   1,1   2,2   3,3   4,4   3,4
 
i8 : degree JnoSat==degree J

o8 = true

See also

Functions with optional argument named Saturate:

Further information

  • Default value: true
  • Function: scoreEquations -- score equations of the log-likelihood function of a Gaussian graphical model
  • Option key: Saturate -- optional input whether to saturate

The source of this document is in GraphicalModelsMLE.m2:1202:0.