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<div><h1>Markov -- Markov ideals, arising from Bayesian networks in statistics</h1>
<div class="single"><h2>Description</h2>
<div><p>This package is used to construct ideals corresponding to discrete graphical models, as described in several places, including the paper: Garcia, Stillman and Sturmfels, "The algebraic geometry of Bayesian networks", J. Symbolic Comput., 39(3-4):331–355, 2005.</p>
<p>The paper also constructs Gaussian ideals, as described in the paper by Seth Sullivant: "Algebraic geometry of Gaussian Bayesian networks", Adv. in Appl. Math. 40 (2008), no. 4, 482--513.</p>
<div>Here is a typical use of this package. We create the ideal in 16 variables whose zero set represents the probability distributions on four binary random variables which satisfy the conditional independence statements coming from the "diamond" graph 4 --> 2,3 --> 1.</div>
<table class="examples"><tr><td><pre>i1 : R = markovRing(2,2,2,2)
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : G = makeGraph{{},{1},{1},{2,3}}
o2 = Graph{1 => set {} }
2 => set {1}
3 => set {1}
4 => set {2, 3}
o2 : Graph</pre>
</td></tr>
<tr><td><pre>i3 : S = globalMarkovStmts G
o3 = {{{1}, {4}, {2, 3}}, {{2}, {3}, {4}}}
o3 : List</pre>
</td></tr>
<tr><td><pre>i4 : I = markovIdeal(R,S)
o4 = ideal (- p p + p p , - p p +
1,1,1,2 2,1,1,1 1,1,1,1 2,1,1,2 1,1,2,2 2,1,2,1
------------------------------------------------------------------------
p p , - p p + p p , -
1,1,2,1 2,1,2,2 1,2,1,2 2,2,1,1 1,2,1,1 2,2,1,2
------------------------------------------------------------------------
p p + p p , - p p +
1,2,2,2 2,2,2,1 1,2,2,1 2,2,2,2 1,1,2,1 1,2,1,1
------------------------------------------------------------------------
p p + p p - p p -
1,1,1,1 1,2,2,1 1,2,2,1 2,1,1,1 1,2,1,1 2,1,2,1
------------------------------------------------------------------------
p p - p p + p p +
1,1,2,1 2,2,1,1 2,1,2,1 2,2,1,1 1,1,1,1 2,2,2,1
------------------------------------------------------------------------
p p , - p p + p p +
2,1,1,1 2,2,2,1 1,1,2,2 1,2,1,2 1,1,1,2 1,2,2,2
------------------------------------------------------------------------
p p - p p - p p -
1,2,2,2 2,1,1,2 1,2,1,2 2,1,2,2 1,1,2,2 2,2,1,2
------------------------------------------------------------------------
p p + p p + p p )
2,1,2,2 2,2,1,2 1,1,1,2 2,2,2,2 2,1,1,2 2,2,2,2
o4 : Ideal of R</pre>
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<div>Sometime an ideal can be simplified by changing variables. Very often, by using <tt>marginMap</tt> (missing documentation<!-- tag: marginMap -->), such ideals can be transformed to binomial ideals. This is the case here.</div>
<table class="examples"><tr><td><pre>i5 : F = marginMap(1,R)
o5 = map(R,R,{p - p , p - p , p - p , p - p , p - p , p - p , p - p , p - p , p , p , p , p , p , p , p , p })
1,1,1,1 2,1,1,1 1,1,1,2 2,1,1,2 1,1,2,1 2,1,2,1 1,1,2,2 2,1,2,2 1,2,1,1 2,2,1,1 1,2,1,2 2,2,1,2 1,2,2,1 2,2,2,1 1,2,2,2 2,2,2,2 2,1,1,1 2,1,1,2 2,1,2,1 2,1,2,2 2,2,1,1 2,2,1,2 2,2,2,1 2,2,2,2
o5 : RingMap R <--- R</pre>
</td></tr>
<tr><td><pre>i6 : J = F I;
o6 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i7 : netList pack(2,J_*)
+-------------------------------------+-------------------------------------+
o7 = |- p p + p p |- p p + p p |
| 1,1,1,2 2,1,1,1 1,1,1,1 2,1,1,2| 1,1,2,2 2,1,2,1 1,1,2,1 2,1,2,2|
+-------------------------------------+-------------------------------------+
|- p p + p p |- p p + p p |
| 1,2,1,2 2,2,1,1 1,2,1,1 2,2,1,2| 1,2,2,2 2,2,2,1 1,2,2,1 2,2,2,2|
+-------------------------------------+-------------------------------------+
|- p p + p p |- p p + p p |
| 1,1,2,1 1,2,1,1 1,1,1,1 1,2,2,1| 1,1,2,2 1,2,1,2 1,1,1,2 1,2,2,2|
+-------------------------------------+-------------------------------------+</pre>
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<div>This ideal has 5 primary components. The first is the one that has statistical significance. The significance of the other components is still poorly understood.</div>
<table class="examples"><tr><td><pre>i8 : time netList primaryDecomposition J
-- used 1.19282 seconds
+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
o8 = |ideal (p p - p p , p p - p p , p p - p p , p p - p p , p p - p p , p p - p p , p p p p - p p p p , p p p p - p p p p , p p p p - p p p p , p p p p - p p p p , p p p p - p p p p , p p p p - p p p p , p p p p - p p p p , p p p p - p p p p , p p p p - p p p p )|
| 1,2,2,2 2,2,2,1 1,2,2,1 2,2,2,2 1,2,1,2 2,2,1,1 1,2,1,1 2,2,1,2 1,1,2,2 2,1,2,1 1,1,2,1 2,1,2,2 1,1,1,2 2,1,1,1 1,1,1,1 2,1,1,2 1,1,2,2 1,2,1,2 1,1,1,2 1,2,2,2 1,1,2,1 1,2,1,1 1,1,1,1 1,2,2,1 2,1,1,1 2,1,2,2 2,2,1,2 2,2,2,1 2,1,1,2 2,1,2,1 2,2,1,1 2,2,2,2 1,1,1,1 2,1,2,2 2,2,1,2 2,2,2,1 1,1,1,2 2,1,2,1 2,2,1,1 2,2,2,2 1,1,2,2 2,1,1,1 2,2,1,2 2,2,2,1 1,1,2,1 2,1,1,2 2,2,1,1 2,2,2,2 1,1,1,1 1,1,2,2 2,2,1,2 2,2,2,1 1,1,1,2 1,1,2,1 2,2,1,1 2,2,2,2 1,2,1,2 2,1,1,1 2,1,2,2 2,2,2,1 1,2,1,1 2,1,1,2 2,1,2,1 2,2,2,2 1,1,1,1 1,2,1,2 2,1,2,2 2,2,2,1 1,1,1,2 1,2,1,1 2,1,2,1 2,2,2,2 1,2,2,2 2,1,1,2 2,1,2,1 2,2,1,1 1,2,2,1 2,1,1,1 2,1,2,2 2,2,1,2 1,1,2,1 1,2,2,2 2,1,1,2 2,2,1,1 1,1,2,2 1,2,2,1 2,1,1,1 2,2,1,2 1,2,1,1 1,2,2,2 2,1,1,2 2,1,2,1 1,2,1,2 1,2,2,1 2,1,1,1 2,1,2,2 |
+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|ideal (p , p , p , p , p p - p p , p p - p p ) |
| 1,2,2,2 1,2,2,1 1,2,1,2 1,2,1,1 1,1,2,2 2,1,2,1 1,1,2,1 2,1,2,2 1,1,1,2 2,1,1,1 1,1,1,1 2,1,1,2 |
+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|ideal (p , p , p , p , p p - p p , p p - p p ) |
| 1,2,2,2 1,2,2,1 1,1,2,2 1,1,2,1 1,2,1,2 2,2,1,1 1,2,1,1 2,2,1,2 1,1,1,2 2,1,1,1 1,1,1,1 2,1,1,2 |
+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|ideal (p , p , p , p , p p - p p , p p - p p ) |
| 1,1,2,2 1,1,2,1 1,1,1,2 1,1,1,1 1,2,2,2 2,2,2,1 1,2,2,1 2,2,2,2 1,2,1,2 2,2,1,1 1,2,1,1 2,2,1,2 |
+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|ideal (p , p , p , p , p p - p p , p p - p p ) |
| 1,2,1,2 1,2,1,1 1,1,1,2 1,1,1,1 1,2,2,2 2,2,2,1 1,2,2,1 2,2,2,2 1,1,2,2 2,1,2,1 1,1,2,1 2,1,2,2 |
+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>The parts of the package involving graphs might eventually be changed to use a package dealing specifically with graphs. This might change the interface to this package.</div>
</div>
</div>
<div class="single"><h2>Authors</h2>
<ul><li><div class="single">Luis Garcia</div>
</li>
<li><div class="single">Mike Stillman</div>
</li>
</ul>
</div>
<div class="single"><h2>Version</h2>
This documentation describes version <b>1.2</b> of Markov.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/Markov.m2">Markov.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<ul><li><span><tt>Graph</tt> (missing documentation<!-- tag: Graph -->)</span></li>
</ul>
</div>
</li>
<li><div class="single">Functions<ul><li><span><tt>displayGraph</tt> (missing documentation<!-- tag: displayGraph -->)</span></li>
<li><span><a href="_gauss__Ideal.html" title="correlation ideal of a Bayesian network of joint Gaussian variables">gaussIdeal</a> -- correlation ideal of a Bayesian network of joint Gaussian variables</span></li>
<li><span><tt>gaussMinors</tt> (missing documentation<!-- tag: gaussMinors -->)</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>gaussTrekIdeal</tt> (missing documentation<!-- tag: gaussTrekIdeal -->)</span></li>
<li><span><tt>globalMarkovStmts</tt> (missing documentation<!-- tag: globalMarkovStmts -->)</span></li>
<li><span><tt>hideMap</tt> (missing documentation<!-- tag: hideMap -->)</span></li>
<li><span><tt>localMarkovStmts</tt> (missing documentation<!-- tag: localMarkovStmts -->)</span></li>
<li><span><tt>makeGraph</tt> (missing documentation<!-- tag: makeGraph -->)</span></li>
<li><span><tt>marginMap</tt> (missing documentation<!-- tag: marginMap -->)</span></li>
<li><span><tt>markovIdeal</tt> (missing documentation<!-- tag: markovIdeal -->)</span></li>
<li><span><tt>markovMatrices</tt> (missing documentation<!-- tag: markovMatrices -->)</span></li>
<li><span><a href="_markov__Ring.html" title="ring of probability distributions on several discrete random variables">markovRing</a> -- ring of probability distributions on several discrete random variables</span></li>
<li><span><tt>pairMarkovStmts</tt> (missing documentation<!-- tag: pairMarkovStmts -->)</span></li>
<li><span><tt>removeRedundants</tt> (missing documentation<!-- tag: removeRedundants -->)</span></li>
<li><span><tt>writeDotFile</tt> (missing documentation<!-- tag: writeDotFile -->)</span></li>
</ul>
</div>
</li>
<li><div class="single">Methods<ul><li><span><tt>displayGraph(String,Graph)</tt> (missing documentation<!-- tag: (displayGraph,String,Graph) -->)</span></li>
<li><span><tt>displayGraph(String,String,Graph)</tt> (missing documentation<!-- tag: (displayGraph,String,String,Graph) -->)</span></li>
<li><span>gaussIdeal(Ring,Graph), see <span><a href="_gauss__Ideal.html" title="correlation ideal of a Bayesian network of joint Gaussian variables">gaussIdeal</a> -- correlation ideal of a Bayesian network of joint Gaussian variables</span></span></li>
<li><span><tt>gaussTrekIdeal(Ring,Graph)</tt> (missing documentation<!-- tag: (gaussTrekIdeal,Ring,Graph) -->)</span></li>
<li><span><tt>writeDotFile(String,Graph)</tt> (missing documentation<!-- tag: (writeDotFile,String,Graph) -->)</span></li>
</ul>
</div>
</li>
<li><div class="single">Other things<ul><li><span><tt>dotBinary</tt> (missing documentation<!-- tag: dotBinary -->)</span></li>
<li><span><tt>jpgViewer</tt> (missing documentation<!-- tag: jpgViewer -->)</span></li>
</ul>
</div>
</li>
</ul>
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