I'd need help in understanding some concepts presented on a paper that I've started to read recently.
The general setting is about seeing graph structures from a probabilistic persepctive, in particular:
Let $\mathcal{G} = (V,\mathcal{E})$ be a graph with a multi-variate random variable $\textbf{X}_i$ associated to each node $i \in V$, and our goal is trying to infer some latent variables $\textbf{Z}_i$ given the initial observed variables $\textbf{X}_i$. The graph is said to be defining a Markov Random Field, writing
\begin{equation} p(\{\textbf{z}_i\}, \{\textbf{x}_i\}) \propto \prod_{i \in V}\Phi(\textbf{x}_i,\textbf{z}_i)\prod_{(i,j) \in \mathcal{E}}\Psi(\textbf{z}_i,\textbf{z}_j) \end{equation}
where I have used the shorthand $\{\textbf{x}_i\}$ for $\{\textbf{x}_i \, , \forall i \in V\}$, and $\Phi,\,\Psi$ are so-called non-negative Potential Functions.
It is also stated that $\Phi(\textbf{x}_i, \textbf{z}_i)$ works as the 'likelihood' of a node feature vector $\textbf{x}_i$ given its latent representation $\textbf{z}_i$, while $\Psi$ controls the dependency between connected nodes.
I just wanted to ask if any one could explain me a little better the roles of these functions $\Phi, \Psi$, why the first would act as a likelihood function, and also the overall perspective lying at the bottom of the above equation that gives a factorization of the joint distribution.
Thanks!
James
Informally, there are two sources of information for $z$, the measurements $x$ and the "opinions" of the neighbors. Given the true value of $z$ we may have a model of the probability to see a measurement $x$. This model can be also interpreted in the other direction: given a measurement $x$, what can be said about the true value $z$. This is called the likelihood to stress that it is not a probability. The $\Phi$ function weights $z$ given $x$, so it acts like a likelihood.