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In the hidden Markov models considered above, the state space of the hidden variables is discrete, while the observations themselves can either be discrete (typically generated from a [[categorical distribution]]) or continuous (typically from a [[Gaussian distribution]]). Hidden Markov models can also be generalized to allow continuous state spaces. Examples of such models are those where the Markov process over hidden variables is a [[linear dynamical system]], with a linear relationship among related variables and where all hidden and observed variables follow a [[Gaussian distribution]]. In simple cases, such as the linear dynamical system just mentioned, exact inference is tractable (in this case, using the [[Kalman filter]]); however, in general, exact inference in HMMs with continuous latent variables is infeasible, and approximate methods must be used, such as the [[extended Kalman filter]] or the [[particle filter]].
Nowadays, inference in hidden Markov models is performed in [[Nonparametric statistics|nonparametric]] settings, where the dependency structure enables [[identifiability]] of the model<ref>{{Cite journal |
=== Bayesian modeling of the transitions probabilities ===
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