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Figure 1. Probabilistic parameters of a hidden Markov model (example) X — states y — possible observations a — state transition probabilities b — output probabilities. In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). [7]
For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model [2] and the Abstract Hidden Markov Model. [3]
There exists a range of different model classes and methodology that make use of latent variables and allow inference in the presence of latent variables. Models include: linear mixed-effects models and nonlinear mixed-effects models; Hidden Markov models; Factor analysis; Item response theory; Analysis and inference methods include:
Markov logic network; Markov model; Markov network; Markov process; Markov property; Markov random field; Markov renewal process; Markov's inequality; Markovian arrival processes; Marsaglia polar method; Martingale (probability theory) Martingale difference sequence; Martingale representation theorem; Master equation; Matched filter; Matching ...
The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. [1] An example of a model for such a field is the Ising model. A discrete ...
In this example, predictions for the weather on more distant days change less and less on each subsequent day and tend towards a steady state vector. [5] This vector represents the probabilities of sunny and rainy weather on all days, and is independent of the initial weather. [5] The steady state vector is defined as:
The hierarchical hidden Markov model (HHMM) is a statistical model derived from the hidden Markov model (HMM). In an HHMM, each state is considered to be a self-contained probabilistic model. More precisely, each state of the HHMM is itself an HHMM. HHMMs and HMMs are useful in many fields, including pattern recognition. [1] [2]
An illustrative example is a bistable system that can be characterized by a hidden Markov model (HMM) subject to measurement noise. Such models are employed for many biological systems: They have, for example, been used in development, cell signaling, activation/deactivation, logical processing and non-equilibrium thermodynamics.