Hidden Markov Random Field Model

Let $\mathcal L$ and $\mathcal D$ be two alphabets:

$${\mathcal L}=\{1, 2, \cdots, l\},\quad {\mathcal D}=\{1, 2, \cdots, d\}.$$

Let ${\mathcal S}=\{1,2,\cdots, N\}$ be the set of indices and $R=\{r_i, i\in \mathcal S\}$ denote any family of random variables indexed by $\mathcal S$, in which each random variable R_i takes a value z_i in its state space. Such a family R is called a random field. The joint event $(R_i=r_i, \cdots, R_N=r_N)$ is simplified to R=r where $r=\{r_1, \cdots, r_N\}$ is a configuration of R, corresponding to a realization of this random field. Let X and Y be two such random fields whose state spaces are $\mathcal L$ and $\mathcal D$ respectively so that for $\forall i \in \mathcal S$ we have $X_i \in \mathcal L$ and $Y_i \in \mathcal D$. Let x denote a configuration of X and $\mathcal X$ be the set of all possible configurations so that

$${\mathcal X}=\{{\mathbf x}=(x_1,\cdots, x_N)\vert x_i \in {\mathcal L}, i \in \mathcal S\}.$$

Similarly, let y be a configuration of Y and $\mathcal Y$ be the set of all possible configurations so that

$${\mathcal Y} =\{{\mathbf y}=(y_1,\cdots, y_N)\vert y_i \in {\mathcal D}, i \in \mathcal S\}.$$

Given $X_i=\ell$, Y_i follows a conditional probability distribution

$$p(y_i\vert\ell) = f(y_i;\theta_\ell), \quad \forall \ell \in \mathcal L$$ (1)

where $\theta_\ell$ is the set of parameters. For all $\ell$, the function family $f(\cdot;\theta_\ell)$ has the same known analytic form. We also assume that (X, Y) is pairwise independent, meaning

$$P({\mathbf y, x})=\prod_{i \in {\mathcal S}}P(y_i, x_i)$$ (2)

In order to develop the HMRF model we firstly take the standard FM model as a comparison.