# scoreEquations -- score equations of the log-likelihood function of a Gaussian graphical model

## Synopsis

• Usage:
scoreEquations(R,U)
• Inputs:
• Optional inputs:
• CovarianceMatrix => ..., default value false, output covariance matrix
• DoSaturate => ..., default value true, remove saturation
• RealPrecision => ..., default value 53, number of decimals used to round input data in RR to data in QQ
• SampleData => ..., default value true, input sample covariance matrix instead of sample data
• SaturateOptions => ..., default value new OptionTable from {Strategy => null, BasisElementLimit => infinity, DegreeLimit => {}, MinimalGenerators => true, PairLimit => infinity}, use options from "saturate"
• Outputs:
• , consisting either of (Ideal) or (Ideal,Matrix) where the ideal is generated by the score equations of the log-likelihood function of the Gaussian model and the matrix is the symbolic covariance matrix of the model

## Description

This function computes the score equations that arise from taking partial derivatives of the log-likelihood function of the concentration matrix (the inverse of the covariance matrix) of a Gaussian graphical statistical model and returns the ideal generated by such equations.

The input of this function is a gaussianRing and statistical data. The latter can be given as a matrix or a list of observations. The rows of the matrix or the elements of the list are observation vectors given as lists. It is possible to input the sample covariance matrix directly by using the optional input SampleData.

 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 : JU=scoreEquations(R,U) o4 = 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 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 : V = sampleCovarianceMatrix U o5 = | 95819/19600 25601/3360 -2129/4480 -1313/16800 | | 25601/3360 867/64 -2321/2304 -173/192 | | -2129/4480 -2321/2304 337/3072 473/11520 | | -1313/16800 -173/192 473/11520 3641/4800 | 4 4 o5 : Matrix QQ <--- QQ i6 : JV=scoreEquations(R,V,SampleData=>false) o6 = 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 o6 : 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

SaturateOptions allows to use all functionalities of saturate. DoSaturate removes the saturation procedure. Note that the latter will not provide the score equations of the model.

 i7 : G = mixedGraph(digraph {{1,2},{1,3},{2,3},{3,4}},bigraph{{3,4}}); i8 : R = gaussianRing(G); i9 : 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 o9 : Matrix QQ <--- QQ i10 : J=scoreEquations(R,U,SaturateOptions => {Strategy => Eliminate}) o10 = 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 o10 : 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 i11 : JnoSat=scoreEquations(R,U,DoSaturate=>false) o11 = ideal (- 574914l + 896035, - 2299656l p - 3584140l p - 1,2 1,3 4,4 2,3 4,4 ----------------------------------------------------------------------- 223545l p - 223545p + 36764p , - 614424l p - 3,4 3,4 4,4 3,4 1,3 4,4 ----------------------------------------------------------------------- 1092420l p - 81235l p - 81235p + 72660p , - 2,3 4,4 3,4 3,4 4,4 3,4 ----------------------------------------------------------------------- 35385l p - 153288l p - 324940l p + 13244p - 3,4 3,3 1,3 3,4 2,3 3,4 3,3 ----------------------------------------------------------------------- 2 35385p , - 19600p + 95819, 1149828l - 3584140l - 235200p 3,4 1,1 1,2 1,2 2,2 ----------------------------------------------------------------------- 2 2 2 2 2 + 3186225, 55191744l p + 172038720l l p + 152938800l p 1,3 4,4 1,3 2,3 4,4 2,3 4,4 ----------------------------------------------------------------------- 2 2 + 10730160l l p p + 22745800l l p p + 1238475l p 1,3 3,4 4,4 3,4 2,3 3,4 4,4 3,4 3,4 3,4 ----------------------------------------------------------------------- 2 2 2 + 10730160l p + 22745800l p - 11289600p p - 1,3 4,4 2,3 4,4 3,3 4,4 ----------------------------------------------------------------------- 1764672l p p - 20344800l p p + 2476950l p p - 1,3 4,4 3,4 2,3 4,4 3,4 3,4 4,4 3,4 ----------------------------------------------------------------------- 2 2 2 927080l p + 11289600p p + 1238475p - 927080p p + 3,4 3,4 4,4 3,4 4,4 4,4 3,4 ----------------------------------------------------------------------- 2 2 2 8563632p , 1238475l p + 10730160l l p p + 3,4 3,4 3,3 1,3 3,4 3,3 3,4 ----------------------------------------------------------------------- 2 2 2 22745800l l p p + 55191744l p + 172038720l l p + 2,3 3,4 3,3 3,4 1,3 3,4 1,3 2,3 3,4 ----------------------------------------------------------------------- 2 2 2 2 152938800l p - 927080l p - 11289600p p - 2,3 3,4 3,4 3,3 3,3 4,4 ----------------------------------------------------------------------- 1764672l p p - 20344800l p p + 2476950l p p + 1,3 3,3 3,4 2,3 3,3 3,4 3,4 3,3 3,4 ----------------------------------------------------------------------- 2 2 2 2 10730160l p + 22745800l p + 11289600p p + 8563632p - 1,3 3,4 2,3 3,4 3,3 3,4 3,3 ----------------------------------------------------------------------- 2 927080p p + 1238475p , - 5365080l l p p - 3,3 3,4 3,4 1,3 3,4 3,3 4,4 ----------------------------------------------------------------------- 2 2 11372900l l p p - 1238475l p p - 55191744l p p - 2,3 3,4 3,3 4,4 3,4 3,3 3,4 1,3 4,4 3,4 ----------------------------------------------------------------------- 2 2 172038720l l p p - 152938800l p p - 5365080l l p 1,3 2,3 4,4 3,4 2,3 4,4 3,4 1,3 3,4 3,4 ----------------------------------------------------------------------- 2 - 11372900l l p + 882336l p p + 10172400l p p - 2,3 3,4 3,4 1,3 3,3 4,4 2,3 3,3 4,4 ----------------------------------------------------------------------- 1238475l p p + 927080l p p - 10730160l p p - 3,4 3,3 4,4 3,4 3,3 3,4 1,3 4,4 3,4 ----------------------------------------------------------------------- 2 22745800l p p + 11289600p p p + 882336l p + 2,3 4,4 3,4 3,3 4,4 3,4 1,3 3,4 ----------------------------------------------------------------------- 2 2 3 10172400l p - 1238475l p - 11289600p + 463540p p - 2,3 3,4 3,4 3,4 3,4 3,3 4,4 ----------------------------------------------------------------------- 2 8563632p p - 1238475p p + 463540p ) 3,3 3,4 4,4 3,4 3,4 o11 : 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

The ML-degree of the model is the degree of the score equations ideal. The ML-degree of the running example is 1:

 i12 : G = mixedGraph(digraph {{1,2},{1,3},{2,3},{3,4}},bigraph{{3,4}}); i13 : R = gaussianRing(G); i14 : 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 o14 : Matrix QQ <--- QQ i15 : J = scoreEquations(R,U) o15 = 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 o15 : 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 i16 : dim J, degree J o16 = (0, 1) o16 : Sequence

## Ways to use scoreEquations :

• "scoreEquations(List,Ring)"
• "scoreEquations(Matrix,Ring)"
• "scoreEquations(Ring,List)"
• "scoreEquations(Ring,Matrix)"

## For the programmer

The object scoreEquations is .