Thermodynamics of nanostructures

A hidden Markov random field is a generalization of a hidden Markov model. Instead of having an underlying Markov chain, hidden Markov random fields have an underlying Markov random field.

Suppose that we observe a random variable $Y_{i}$, where $i\in S$. Hidden Markov random fields assume that the probabilistic nature of $Y_{i}$ is determined by the unobservable Markov random field $X_{i}$, $i\in S$. That is, given the neighbors $N_{i}$ of $X_{i}$, $X_{i}$ is independent of all other $X_{j}$ (Markov property). The main difference with a hidden Markov model is that neighborhood is not defined in 1 dimension but within a network, i.e. $X_{i}$ is allowed to have more than the two neighbors that it would have in a Markov chain. The model is formulated in such a way that given $X_{i}$, $Y_{i}$ are independent (conditional independence of the observable variables given the Markov random field).

See also

• Hidden Markov model
• Markov network
• Bayesian network

References

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