<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
    <meta content="text/html; charset=ISO-8859-1"
          http-equiv="content-type">
    <title>Update</title>
</head>
<body>
<table width="95%" border="1" bgcolor="maroon">
    <tr>
        <td><h2><font color="#FFFFFF">Inside the Update Box</font></h2></td>
    </tr>
</table>
<p>An Instantiated Model box in the main workspace looks like this:</p>
<p><img src="../../images/im_highlight.gif" width="276" height="67"></p>
<p><font size="+1"><img
        src="../../images/Untitledupdaterbox.jpg" alt="" width="250" height="94"
        style="width: 250px; height: 94px;"></font></p>
<p><br style="font-family: times new roman;">
    The functions inside the Update Box enable you to use an Instantiated Bayes
    Net model to compute the conditional probability of any variable in the model
    from values you specify for any other variables in the model.<br
            style="font-family: times new roman;">
    <br style="font-family: times new roman;">
    <span style="font-family: times new roman;">Tetrad has three programs for updatin</span>g:
    (1) Approximate Updater, (2) Row Summing Exact Updater, and (3) CPT Invariant
    Updater. <br>
</p>
<h3>1. Approximate Updater </h3>
<p>Calculates updated marginals for a Bayes net by simulating data and calculating
    likelihood ratios. The method is as follows. For P(A | E) (where E is the
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence),
    enough sample points are simulated from the underlying Bayes Im so that 1000
    satisfy the condition E, keeping track of the number n that satisfy condition
    A. Then the maximum likelihood estimate of P(A | E) is calculated as n / 1000.
    <br>
    <br>
    The approximate updater runs quite quickly, even for large numbers of variables,
    so long as the number of variables in
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence
    is small. The more variables
    there are in
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence,
    the more sample points need to be generated to achieve
    1000 samples points satisfying E. <br>
</p>
<h3>2. Row Summing Exact Updater </h3>
<p>Calculates updated marginals P(A | E) for a Bayes net (where E is the
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence)
    by summing probabilities for rows in the joint probability table that satisfy
    condition E, summing probabilities for for rows in the joint probability table
    that satisfy condition A &amp; E, and dividing the second sum by the first.
    A row in the joint probability in this sense is a combination of values for
    the variables of the Bayes net mapped to the probability of that combination
    of values occurring in a sample. This probability is calculated for each row
    from the conditional probability tables of the Bayes net using the standard
    factorization of the Bayes net. <br>
    <br>
    The row summing updater can be extremely expensive if the number of variables
    in the Bayes net is large and the number of variables in
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence
    is small.
    However, the row summing updater can be extremely useful (and fast) if almost
    all of the variables in the Bayes net are in
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence.
    For instance, if all
    but one variable (say, X) is in
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence,
    then the number of rows in the joint
    probability table that have to be examined in order to calculate marginals for
    X is just the number of categories of X. <br>
</p>
<h3>3. CPT Invariant Updater </h3>
<p>Calculates updated marginals P(A | E) for a Bayes net (where E is the
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence)
    by breaking the problem down into two parts: first, calculating an "updated
    Bayes net" (in a sense to be defined), and second, calculating marginals recursively
    from this updated Bayes net. Probabilities for a Bayes net are specified in
    terms of conditional probability tables for its variables. These are tables
    of the probability for each category of a variable conditional on each combination
    of parent values of that variable, P(V = v' | P1 = p1' &amp; ...&amp; Pn = pn').
    Define an "updated Bayes net" as the Bayes net in which each of these probabilities
    has been replaced by P(V = v' | P1 = p1' &amp;... &amp; Pn = pn' &amp; E). (These
    replacement values will not be defined if the conjunction P1 = p1' &amp;...
    &amp; Pn = pn' &amp; E is impossible.) It is straightforward to show that marginals
    for such an updated Bayes net just are the updated marginals for the original
    Bayes net. <br>
    <br>
    In updating a Bayes net, in the sense defined above, only the conditional probabilty
    tables for ancestors of the
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence
    variables are altered. This suggests an
    algorithm for updating a Bayes net given
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence
    E. For each variable that's
    an ancestor of
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence,
    use the row summing method to calculate updated conditional
    probabilities in that variable's conditional probability table. Otherwise, just
    keep the conditional probabilities from the original Bayes net. This is the
    algorithms that is implemented. <br>
    <br>
    For calculating single-variable marginals from a Bayes net, Bayes Theorem is
    used recursively. For example, if X--&gt;Y, where X and Y both have categories
    {0, 1}, P(Y = 0) = P(Y = 0 | X = 0) P(X = 0) + P(Y = 0 | X = 1) P(X = 1). Since
    all of the probabilites on the right side of this equation are stored in the
    conditional probability tables for the Bayes net, this values can be calculated
    directly. For longer chains, more recursion would have to be done to calculate
    marginals for intervening variables. These intervening marginals can, however,
    once calculated, be stored for later use, and they are. <br>
    <br>
    The algorithm slows down (as do most updating algorithms) when dealing with
    graphs where parents of a modelNode are d-connected (see Spirtes, et al 2000 for
    the exact definition). For instance, in this graph: <br>
    <br>
    &nbsp;&nbsp; X--&gt;Y <br>
    &nbsp;&nbsp; X--&gt;Z <br>
    &nbsp;&nbsp; Y--&gt;W <br>
    &nbsp;&nbsp; Z--&gt;W <br>
    &nbsp;&nbsp; W--&gt;R <br>
    <br>
    calculating R requires much more extensive calculation than for this graph:
    <br>
    <br>
    &nbsp;&nbsp; X1--&gt;Y <br>
    &nbsp;&nbsp; X2--&gt;Z <br>
    &nbsp;&nbsp; Y--&gt;W <br>
    &nbsp;&nbsp; Z--&gt;W <br>
    &nbsp;&nbsp; W--&gt;R.&nbsp;</p>
<p>In this case, in order to calculate marginals for W, one needs to know probabilities
    of W given particular combinations of parent values of W, which are given in
    the conditional probability tables of the Bayes net, and one also needs to know
    the probabilities of the various combinations of parent values occurreing. For
    example, say that Y, Z have categories {0, 1} in the first graph, above, and
    one wants to know the probability P(W = 0). One can calculate this probability
    as P(W = 0 | Y = 0, Z = 0) P(Y = 0, Z = 0) + P(W = 0 | Y = 0, Z = 1) P(Y = 0,
    Z = 1) + P(W = 0 | Y = 1, Z = 0) P(Y = 1, Z = 0) + P(W = 0 | Y = 1, Z = 1) P(Y
    = 1, Z = 1). The problem with d-connected parents is calculating, e.g., P(Y
    = 0, Z = 0). The CPT invariant updater calculates this probability in a standard
    way, as P(Y = 0) P(Z = 0 | Y = 0). This requires a recursive application of
    the marginal calculating procedure and is expensive. However, the problem of
    d-connected parents of variables is a standard problem (even if not always phrased
    that way) for updating procedures. <br>
    <br>
    In general, the CPT invariant updater is quite fast, but can be slowed down
    for two reasons: (a) the subgraph of the Bayes net restricted to ancestors of
    manipulationEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidencetionEvidence
    variables is complicated, forcing more updated conditional probabilities
    to be calculated, and (b) there are a lot of variables in the Bayes net whose
    parents are moderately or strongly d-connected. </p>
<p>Types of updaters:</p>
<ul>
    <li><a href="row_summing_updater.html">Row Summing Updater</a></li>
    <li><a href="cpt_invariant_updater.html">CPT Invariant Updater</a></li>
    <li><a href="approximate_updater.html">Approximate Updater </a><br>
    </li>
</ul>
</body>
</html>
