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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]
The forward–backward algorithm is an inference algorithm for hidden Markov models which computes the posterior marginals of all hidden state variables given a sequence of observations/emissions ::=, …,, i.e. it computes, for all hidden state variables {, …,}, the distribution ( | :).
A hidden Markov model is a Markov chain for which the state is only partially observable or noisily observable. In other words, observations are related to the state of the system, but they are typically insufficient to precisely determine the state. Several well-known algorithms for hidden Markov models exist.
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events. This is done especially in the context of Markov information sources and hidden Markov models (HMM).
Russell and Norvig's Artificial Intelligence, a Modern Approach, starting on page 570 of the 2010 edition, provides a succinct exposition of this and related topics; Smyth, Padhraic, David Heckerman, and Michael I. Jordan. "Probabilistic independence networks for hidden Markov probability models." Neural computation 9.2 (1997): 227-269.
A hidden Markov model describes the joint probability of a collection of "hidden" and observed discrete random variables.It relies on the assumption that the i-th hidden variable given the (i − 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state.
Layered hidden Markov model This page was last edited on 30 March 2013, at 04:46 (UTC). Text is available under the Creative Commons Attribution-ShareAlike 4.0 ...
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]