In statistics, 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 ${\displaystyle Y_{i}}$, where ${\displaystyle i\in S}$. Hidden Markov random fields assume that the probabilistic nature of ${\displaystyle Y_{i}}$ is determined by the unobservable Markov random field ${\displaystyle X_{i}}$, ${\displaystyle i\in S}$. That is, given the neighbors ${\displaystyle N_{i}}$ of ${\displaystyle X_{i},X_{i}}$ is independent of all other ${\displaystyle X_{j}}$ (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. ${\displaystyle X_{i}}$ 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 ${\displaystyle X_{i}}$, ${\displaystyle Y_{i}}$ are independent (conditional independence of the observable variables given the Markov random field).

In the vast majority of the related literature, the number of possible latent states is considered a user-defined constant. However, ideas from nonparametric Bayesian statistics, which allow for data-driven inference of the number of states, have been also recently investigated with success, e.g.^[1]

See also

• Hidden Markov model
• Markov network
• Bayesian network

References

• Yongyue Zhang; Smith, Stephen; Brady, Michael (11 May 2000). "Hidden Markov Random Field Model". Hidden Markov Random Field Model and Segmentation of Brain MR Images. Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB). FMRIB Technical Report TR00YZ1.