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<head><title>gaussIdeal -- correlation ideal of a Bayesian network of joint Gaussian variables</title>
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<div><h1>gaussIdeal -- correlation ideal of a Bayesian network of joint Gaussian variables</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>gaussIdeal(R,G)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, created with <a href="_gauss__Ring.html" title="ring of gaussian correlations on n random variables">gaussRing</a></span></li>
<li><span><tt>G</tt>, <span>an object of class <tt>Graph</tt> (missing documentation<!-- tag: Graph -->)</span> or <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
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</li>
<li><div class="single">Outputs:<ul><li><span>the ideal in R of the relations in the correlations of the random variables implied by the independence statements of the graph G or the list of independence statements G</span></li>
</ul>
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</ul>
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<div class="single"><h2>Description</h2>
<div>These ideals were first written down by Seth Sullivant, in "Algebraic geometry of Gaussian Bayesian networks". The routines in this package involving Gaussian variables are all based on that paper.<table class="examples"><tr><td><pre>i1 : R = gaussRing 5;</pre>
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<tr><td><pre>i2 : G = makeGraph {{2},{3},{4,5},{5},{}}
o2 = Graph{1 => set {2} }
2 => set {3}
3 => set {4, 5}
4 => set {5}
5 => set {}
o2 : Graph</pre>
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<tr><td><pre>i3 : (globalMarkovStmts G)/print;
{{1, 2}, {4, 5}, {3}}
{{1}, {3, 4, 5}, {2}}</pre>
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<tr><td><pre>i4 : J = gaussIdeal(R,G)
o4 = ideal (- s s + s s , - s s + s s , - s s +
1,5 2,4 1,4 2,5 1,5 3,4 1,4 3,5 2,5 3,4
------------------------------------------------------------------------
s s , s s - s s , s s - s s , s s - s s ,
2,4 3,5 1,4 2,3 1,3 2,4 1,4 3,3 1,3 3,4 2,4 3,3 2,3 3,4
------------------------------------------------------------------------
s s - s s , s s - s s , s s - s s , -
1,5 2,3 1,3 2,5 1,5 3,3 1,3 3,5 2,5 3,3 2,3 3,5
------------------------------------------------------------------------
s s + s s , - s s + s s , - s s + s s ,
1,4 2,3 1,3 2,4 1,5 2,3 1,3 2,5 1,5 2,4 1,4 2,5
------------------------------------------------------------------------
s s - s s , s s - s s , s s - s s )
1,3 2,2 1,2 2,3 1,4 2,2 1,2 2,4 1,5 2,2 1,2 2,5
o4 : Ideal of R</pre>
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<p/>
A list of independence statments (as for example returned by globalMarkovStmts) can be provided instead of a graph.<p/>
The ideal corresponding to a conditional independence statement {A,B,C} (where A,B,C, are disjoint lists of integers in the range 1..n (n is the number of random variables) is the #C+1 x #C+1 minors of the submatrix of the generic symmetric matrix M = (s_(i,j)), whose rows are in A union C, and whose columns are in B union C. In general, this does not need to be a prime ideal.<table class="examples"><tr><td><pre>i5 : I = gaussIdeal(R,{{{1,2},{4,5},{3}}, {{1},{2},{3,4,5}}})
o5 = ideal (- s s + s s , - s s + s s , - s s +
1,5 2,4 1,4 2,5 1,5 3,4 1,4 3,5 2,5 3,4
------------------------------------------------------------------------
s s , s s - s s , s s - s s , s s - s s ,
2,4 3,5 1,4 2,3 1,3 2,4 1,4 3,3 1,3 3,4 2,4 3,3 2,3 3,4
------------------------------------------------------------------------
s s - s s , s s - s s , s s - s s ,
1,5 2,3 1,3 2,5 1,5 3,3 1,3 3,5 2,5 3,3 2,3 3,5
------------------------------------------------------------------------
2 2
s s s - s s s s - s s s s + s s s -
1,5 2,5 3,4 1,5 2,4 3,4 3,5 1,4 2,5 3,4 3,5 1,4 2,4 3,5
------------------------------------------------------------------------
2
s s s s + s s s s + s s s s - s s s +
1,5 2,5 3,3 4,4 1,5 2,3 3,5 4,4 1,3 2,5 3,5 4,4 1,2 3,5 4,4
------------------------------------------------------------------------
s s s s + s s s s - s s s s -
1,5 2,4 3,3 4,5 1,4 2,5 3,3 4,5 1,5 2,3 3,4 4,5
------------------------------------------------------------------------
s s s s - s s s s - s s s s +
1,3 2,5 3,4 4,5 1,4 2,3 3,5 4,5 1,3 2,4 3,5 4,5
------------------------------------------------------------------------
2 2
2s s s s + s s s - s s s - s s s s +
1,2 3,4 3,5 4,5 1,3 2,3 4,5 1,2 3,3 4,5 1,4 2,4 3,3 5,5
------------------------------------------------------------------------
2
s s s s + s s s s - s s s - s s s s +
1,4 2,3 3,4 5,5 1,3 2,4 3,4 5,5 1,2 3,4 5,5 1,3 2,3 4,4 5,5
------------------------------------------------------------------------
s s s s )
1,2 3,3 4,4 5,5
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : codim I
o6 = 5</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><tt>makeGraph</tt> (missing documentation<!-- tag: makeGraph -->)</span></li>
<li><span><tt>globalMarkovStmts</tt> (missing documentation<!-- tag: globalMarkovStmts -->)</span></li>
<li><span><tt>localMarkovStmts</tt> (missing documentation<!-- tag: localMarkovStmts -->)</span></li>
<li><span><a href="_gauss__Ring.html" title="ring of gaussian correlations on n random variables">gaussRing</a> -- ring of gaussian correlations on n random variables</span></li>
<li><span><tt>gaussMinors</tt> (missing documentation<!-- tag: gaussMinors -->)</span></li>
<li><span><tt>gaussTrekIdeal</tt> (missing documentation<!-- tag: gaussTrekIdeal -->)</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>gaussIdeal</tt> :</h2>
<ul><li>gaussIdeal(Ring,Graph)</li>
<li>gaussIdeal(Ring,List)</li>
</ul>
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