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Bayesian network

A Bayesian network is a form of probabilistic graphical model, also known as Bayesian belief network or just belief network.
A Bayesian network can be represented by a graph (as in graph theory) with probabilities attached. Thus, a Bayesian network represents a set of variables together with a joint probability distribution with explicit independence assumptions.
  • 1 Definition
  • 2 Example
  • 3 Causal Bayesian networks
  • 4 Structure learning
  • 5 Parameter learning
  • 6 Inference
  • Because a Bayesian network is a complete model for the variables and their relationships, it can be used to answer probabilistic queries about them. For example, the network can be used to find out updated knowledge of the state of a subset of variables when other variables (the evidence variables) are observed. This process of computing the posterior distribution of variables given evidence is called probabilistic inference. The posterior gives a universal sufficient statistic for detection applications, when one wants to choose values for the variable subset which minimize some expected loss function, for instance the probability of decision error. A Bayesian network can thus be considered a mechanism for automatically constructing extensions of Bayes' theorem to more complex problems.
    The most common exact inference methods are variable elimination, which eliminates (by integration or summation) the non-observed non-query variables one by one by distributing the sum over the product; clique tree propagation, which caches the computation so that many variables can be queried at one time and new evidence can be propagated quickly; and recursive conditioning, which allows for a space-time tradeoff and matches the efficiency of variable elimination when enough space is used. All of these methods have complexity that is exponential in the network's treewidth. The most common approximate inference algorithms are stochastic MCMC simulation, mini-bucket elimination which generalizes loopy belief propagation, and variational methods.

  • 7 Applications
  • 8 See also
  • 9 Links and software
  • 10 References