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            <h2><font color="#FFFFFF">Tetrad Graph Types </font></h2>
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<p>The theory of causal search that the Tetrad program implements uses graphs of a variety of different types, some
    simple and some fairly sophisticated. A brief description of each of the main types is given below. For more
    details, consult the Bibliography, especially Spirtes, Glymour and Scheines (2002), <em>Causation, Prediction, and
        Search</em>. </p>
<p>&nbsp;</p>
<h3>Directed Graphs.</h3>
<p>A directed graph is a set of variables V together with a set of directed edges Vi--&gt;Vj for Vi, Vj in V, Vi not
    equal to Vj. A directed graph may contain cycles--that is, paths of the form X--&gt;...--&gt;X, for some X in
    V. </p>
<p>&nbsp;</p>
<h3>Directed Acyclic Graphs (DAG).</h3>
<p>A directed acyclic graph is a directed graph that does not contain cycles. This type of graph can be used to
    construct a Bayes or SEM parametric model. A Bayes parametric model requires a DAG.</p>
<h3>SEM Graph. </h3>
<p>A SEM graph is a directed graph over a set of variables V which has been embellished by adidtional variables E
    representing error terms for endogenous variables in V, edges from each e in E to its corresponding variable in V,
    and a set of bidirected edges over this embellished graph. SEM graphs are used to represent the causal structure of
    SEM models. </p>
<h3>CPDAG (Pattern).</h3>
<p>From Causation, Prediction and Search (2002), p. 61: &quot;A CPDAG Pi is a mixed graph with directed and undirected
    edges. A graph G is in the set of graphs represented by Pi if and only if:</p>
<p>(i) G has the same adjacency relations as Pi;<br>
    (ii) if the edge between A and B is oriented A--&gt;B in Pi, then it is oriented A--&gt;B in G;<br>
    (iii) if Y is an unshielded collider on the path &lt;X, Y, Z&gt; in G then Y is an unshielded collider on &lt;X, Y,
    Z&gt; in Pi.&quot;</p>
<p>Y is an unshielded collider on path &lt;X, Y, Z&gt; iff X--&gt;Y&lt;--Z and X and Z are not adjacent. </p>
<p>CPDAGs are theoretically output by the PC and GES algorithms. Sometimes the PC algorithm includes in its output
    bidirected edges. When this happens, it is because there is conflicting independence test information. This usually
    means that the assumption of causal sufficiency has been violated and that FCI should be run as well for
    comparison. </p>
<h3>POIPG.</h3>
<p>POIPG's (pronounced &quot;poip-G&quot;) were output by earlier versions of FCI; due to theoretical advances since
    then, FCI now outputs PAGs (see below).</p>
<p>&nbsp;</p>
<h3>Partial Ancestral Graph (PAG).</h3>
<p>A PAG is ...</p>
<p>PAGs are output by FCI and CCD. </p>
<p>&nbsp;</p>
<h3><strong>Mixed Ancestral Graphs (MAG).</strong></h3>
<p>Definition. </p>
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