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<!DOCTYPE html> <html lang="en"> <head> <title>Markov -- Markov ideals, arising from Bayesian networks in statistics</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Markov ideals, arising from Bayesian networks in statistics" href="index.html">Markov</a> :: <a title="Markov ideals, arising from Bayesian networks in statistics" href="index.html">Markov</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_display__Graph.html">next</a> | previous | <a href="_display__Graph.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Markov -- Markov ideals, arising from Bayesian networks in statistics</h1> <div> <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> <p>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.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = markovRing(2,2,2,2) o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : G = makeGraph{{},{1},{1},{2,3}} o2 = Graph{1 => set {} } 2 => set {1} 3 => set {1} 4 => set {2, 3} o2 : Graph</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : S = globalMarkovStmts G o3 = {{{1}, {4}, {2, 3}}, {{2}, {3}, {4}}} o3 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> </table> <div> <p>Sometime an ideal can be simplified by changing variables. Very often, by using <span class="tt">marginMap</span> (missing documentation)<!--tag: marginMap--> , such ideals can be transformed to binomial ideals. This is the case here.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : J = F I; o6 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">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| +-------------------------------------+-------------------------------------+</code></pre> </td> </tr> </table> <div> <p>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.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i8 : time netList primaryDecomposition J -- used 2.62344s (cpu); 1.72533s (thread); 0s (gc) +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ o8 = |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 ) | | 1,2,1,2 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 | +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |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 , 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 | +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> <div> <p>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.</p> </div> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li>Luis David Garcia Puente</li> <li>Mike Stillman</li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.3</b> of Markov, released <b>May 15, 2021</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{MarkovSource, title = {{Markov: Markov ideals arising from Bayesian networks in statistics. Version~1.3}}, author = {Luis David Garcia Puente and Mike Stillman}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Types <ul> <li><span><span class="tt">Graph</span> (missing documentation)<!--tag: Graph--> </span></li> </ul> </li> <li>Functions and commands <ul> <li><span><span class="tt">displayGraph</span> (missing documentation)<!--tag: displayGraph--> </span></li> <li><span><a title="correlation ideal of a Bayesian network of joint Gaussian variables" href="_gauss__Ideal.html">gaussIdeal</a> -- correlation ideal of a Bayesian network of joint Gaussian variables</span></li> <li><span><span class="tt">gaussMinors</span> (missing documentation)<!--tag: gaussMinors--> </span></li> <li><span><a title="ring of gaussian correlations on n random variables" href="_gauss__Ring.html">gaussRing</a> -- ring of gaussian correlations on n random variables</span></li> <li><span><span class="tt">gaussTrekIdeal</span> (missing documentation)<!--tag: gaussTrekIdeal--> </span></li> <li><span><span class="tt">globalMarkovStmts</span> (missing documentation)<!--tag: globalMarkovStmts--> </span></li> <li><span><span class="tt">hideMap</span> (missing documentation)<!--tag: hideMap--> </span></li> <li><span><span class="tt">localMarkovStmts</span> (missing documentation)<!--tag: localMarkovStmts--> </span></li> <li><span><span class="tt">makeGraph</span> (missing documentation)<!--tag: makeGraph--> </span></li> <li><span><span class="tt">marginMap</span> (missing documentation)<!--tag: marginMap--> </span></li> <li><span><span class="tt">markovIdeal</span> (missing documentation)<!--tag: markovIdeal--> </span></li> <li><span><span class="tt">markovMatrices</span> (missing documentation)<!--tag: markovMatrices--> </span></li> <li><span><a title="ring of probability distributions on several discrete random variables" href="_markov__Ring.html">markovRing</a> -- ring of probability distributions on several discrete random variables</span></li> <li><span><span class="tt">pairMarkovStmts</span> (missing documentation)<!--tag: pairMarkovStmts--> </span></li> <li><span><span class="tt">removeRedundants</span> (missing documentation)<!--tag: removeRedundants--> </span></li> <li><span><span class="tt">writeDotFile</span> (missing documentation)<!--tag: writeDotFile--> </span></li> </ul> </li> <li>Methods <ul> <li><span><span class="tt">displayGraph(Graph)</span> (missing documentation)<!--tag: (displayGraph,Graph)--> </span></li> <li><span><span class="tt">displayGraph(String,Graph)</span> (missing documentation)<!--tag: (displayGraph,String,Graph)--> </span></li> <li><span><span class="tt">displayGraph(String,String,Graph)</span> (missing documentation)<!--tag: (displayGraph,String,String,Graph)--> </span></li> <li><span><kbd>gaussIdeal(Ring,Graph)</kbd> -- see <span><a title="correlation ideal of a Bayesian network of joint Gaussian variables" href="_gauss__Ideal.html">gaussIdeal</a> -- correlation ideal of a Bayesian network of joint Gaussian variables</span></span></li> <li><span><kbd>gaussIdeal(Ring,List)</kbd> -- see <span><a title="correlation ideal of a Bayesian network of joint Gaussian variables" href="_gauss__Ideal.html">gaussIdeal</a> -- correlation ideal of a Bayesian network of joint Gaussian variables</span></span></li> <li><span><span class="tt">gaussMinors(Matrix,List)</span> (missing documentation)<!--tag: (gaussMinors,Matrix,List)--> </span></li> <li><span><kbd>gaussRing(ZZ)</kbd> -- see <span><a title="ring of gaussian correlations on n random variables" href="_gauss__Ring.html">gaussRing</a> -- ring of gaussian correlations on n random variables</span></span></li> <li><span><span class="tt">gaussTrekIdeal(Ring,Graph)</span> (missing documentation)<!--tag: (gaussTrekIdeal,Ring,Graph)--> </span></li> <li><span><span class="tt">globalMarkovStmts(Graph)</span> (missing documentation)<!--tag: (globalMarkovStmts,Graph)--> </span></li> <li><span><span class="tt">hideMap(ZZ,Ring)</span> (missing documentation)<!--tag: (hideMap,ZZ,Ring)--> </span></li> <li><span><span class="tt">localMarkovStmts(Graph)</span> (missing documentation)<!--tag: (localMarkovStmts,Graph)--> </span></li> <li><span><span class="tt">makeGraph(List)</span> (missing documentation)<!--tag: (makeGraph,List)--> </span></li> <li><span><span class="tt">marginMap(ZZ,Ring)</span> (missing documentation)<!--tag: (marginMap,ZZ,Ring)--> </span></li> <li><span><span class="tt">markovIdeal(Ring,List)</span> (missing documentation)<!--tag: (markovIdeal,Ring,List)--> </span></li> <li><span><span class="tt">markovMatrices(Ring,List)</span> (missing documentation)<!--tag: (markovMatrices,Ring,List)--> </span></li> <li><span><span class="tt">pairMarkovStmts(Graph)</span> (missing documentation)<!--tag: (pairMarkovStmts,Graph)--> </span></li> <li><span><span class="tt">writeDotFile(String,Graph)</span> (missing documentation)<!--tag: (writeDotFile,String,Graph)--> </span></li> </ul> </li> <li>Mutable symbols <ul> <li><span><span class="tt">dotBinary</span> (missing documentation)<!--tag: dotBinary--> </span></li> </ul> </li> <li>Other things <ul> <li><span><span class="tt">dotBinary</span> (missing documentation)<!--tag: dotBinary--> </span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Markov ideals, arising from Bayesian networks in statistics" href="index.html">Markov</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Markov.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Markov.m2:558:0</span>.</p> </div> </div> </div> </body> </html>