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Bayesian network
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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
In order to fully specify the Bayesian network and thus fully represent the joint probability distribution, it is necessary to further specify for each node X the probability distribution for X conditional upon X's parents. The distribution of X conditional upon its parents may have any form. It is common to work with discrete or Gaussian Distributions since that simplifies calculations. Sometimes only constraints on a distribution are known; one can then use the principle of maximum entropy to determine a single distribution, the one with the greatest entropy given the constraints. (Analogously, in the specific context of a dynamic Bayesian network, one commonly specifies the conditional distribution for the hidden state's temporal evolution to maximize the entropy rate of the implied stochastic process.)
Often these conditional distributions include parameters which are unknown and must be estimated from data, sometimes using the maximum likelihood approach. Direct maximization of the likelihood (or of the posterior probability) is often complex when there are unobserved variables. A classical approach to this problem is the expectation-maximization algorithm which alternates computing expected values of the unobserved variables conditional on observed data, with maximizing the complete likelihood (or posterior) assuming that previously computed expected values are correct. Under mild regularity conditions this process converges on maximum likelihood (or maximum posterior) values for parameters. A more fully Bayesian approach to parameters is to treat parameters as additional unobserved variables and to compute a full posterior distribution over all nodes conditional upon observed data, then to integrate out the parameters. This approach can be expensive and lead to large dimension models, so in practise classical parameter-setting approaches are more common
- 6 Inference
- 7 Applications
- 8 See also
- 9 Links and software
- 10 References
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