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I am trying to more deeply understand the difference between Hidden Markov Models and Bayesian Network? The general idea is that HMMs have a single variable which has probabilities of entering different states, known and unknown, whereas the Bayesian network relates multiple random variables.

So, given that, what would examples of particular problems that would be appropriate for each?

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A Hidden Markov Model can be expressed as an instance of a Bayesian network of a particular form. Consequently, a HMM can be viewed as an special case or kind of Bayesian network. Bayesian networks are more general, and can express other kinds of probabilistic structures as well. All HMM's are Bayesian networks, but not all Bayesian networks are HMMs.

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