MRF definition: not all cliques are required to have factors?

Question

I'm reading the notes here. The formal definiton states

A Markov Random Field (MRF) is a probability distribution $p$ over variables $x_1,\ldots,x_n$ defined by an undirected graph $G$ in which nodes correspond to variables $x_i$. The probability $p$ has the form $p(x_1,\ldots,x_n)=\frac{1}{Z} \Pi_{c \in C} \phi_c(x_c)$, where $C$ denotes the set of cliques of $G$.

Question 1: In this notation, what does $x_c$ mean? I'm guessing it's some sort of restriction of the variables in clique $c$ to the values $x_1, \ldots, x_n$?

Question 2: They go on to write:

Note that we do not need to specifiy a factor for each clique.

But the above product runs over all possible cliques, so how does this work? Technically we can specify no factors at all?

Answer 1

Let's take the 4-node example further up on the page you linked to. The set of all cliques is $C = \{\{A\},\{B\},\{C\},\{D\},\{A,B\},\{B,C\},\{C,D\},\{A,D\}\}$.

1. For each clique $c\in C$, $X_c$ represents the set of random variables in that clique. Similarly, $x_c$ is a particular assignment of values to those random variables.

2. $p(a,b,c,d)$ is a valid MRF if and only if it factors over the cliques in the graph, i.e. it can be written in the form $$\begin{align*} p(a,b,c,d) & = \frac{1}{Z} \prod_{c \in C} \phi_c(x_c) \\ & = \frac{1}{Z} \phi_A(a)\phi_B(b) \phi_C(c)\phi_D(d)\phi_{A,B}(a,b)\phi_{B,C}(b,c)\phi_{C,D}(c,d)\phi_{A,D}(a,d). \end{align*} $$ In the example, they only assign "real" factors to the 2-member cliques: $$p(a,b,c,d) = \frac{1}{Z}\phi_{A,B}(a,b)\phi_{B,C}(b,c)\phi_{C,D}(c,d)\phi_{A,D}(a,d).$$ This is what is meant by the fact that not every clique must have a corresponding factor. (Note, however, that every factor must have a corresponding clique in the graph.) To reconcile this with the fact that the product ranges over every clique in $C$, we can simply consider the factors assigned to the singleton cliques $\{A\},\{B\},\{C\},\{D\}$ to be equal to 1. In this view, every clique has a factor, but only some have nontrivial factors.