Inference in a Bayesian Network based on Stochastic Simulation

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Inference in a Bayesian Network based on
Stochastic Simulation

• Assume that we want to compute P(X1TX4T,
  X6T).
• We start with a sequence of input patterns I1,
  I2, , In. Each input pattern specifies a value
  for each root node. For example, I1 X1F,
  X2T, X3T.
• In the input pattern sequence, the frequency of
  each possible random variable value must conforms
  with the prior probability. For example

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• Let us use the pattern X1F, X2T, X3T to
  illustrate the process.
• We start from the root nodes and walk down.
• When the parents of a node are all instantiated,
  the node is ready for being instantiated.
• Since X2T, X3T, the probability that X5T is
  0.1. We can invoke a random number generator to
  give us a number between 0 and 1. If the number
  generated is less than 0.1, then we set X5T.
  Otherwise, we set X5F.

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• Once all simulation runs have been completed, we
  calculate

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Learning in Bayesian Network

• In general, people employ the cause-and-effect
  reasoning to determine the structure of a
  Bayesian network.
• If the training samples contain observations at
  each node, then the learning process is
  straightforward.

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• However, it is common that we do not have
  observations of all the nodes.Therefore, we want
  to find a setting of the conditional probability
  tables so that is maximized, whereis a set of
  training samples.
• We assume that the training samples are collected
  from independent runs of the corresponding
  experiment. Therefore,

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• Since ln(x) is a monotonic increasing function,
  is maximized if and only if is
  maximized.
• For each entry in the conditional probability
  tables that is yet to be determined, we compute

k-th column
where wijk is the entry at row j,column k of the
conditionalprobability table of node Xi.
j-th row
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• Therefore,
• Furthermore,

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• The learning process begins wih an initial guess
  of all unknown wijk.For each unknown wijk,
  computeand set new
• Normalize wijk by setting normalized
• If each wijk converges, then terminate.
  Otherwise, report the process again.