# How maximal clique parameterization obscures original structure

While reading the chapter on Markov networks, I came across the following statement:

Although it can be used without loss of generality, the parameterization using maximal clique potentials generally obscures structure that is present in the original set of factors.

The example given to illustrate above statement is related to the factor for every pair of vertices in a complete undirected Markov graph vs single factor representing the total graph. In the former case, it needs $$4\binom{n}{2}$$ parameters and the latter case contains $$2^{n}-1$$ parameters.

But I didn't understand the sentence given above (bolded) properly. How to understand it properly? How it obscure if we represent factors for maximal cliques in a Markov network?

You are referring to the book Probabilistic Graphical Models by Koller and Frienman, so I'l use the same book as a reference here. The bolded section refers to the part right before it, where the concept of parametrization using maximal cliques is explained. This parametrization means you take each factor of the Gibbs distribution and assign it to a maximal clique that encompasses its scope. This induces a "new" factorization, where the factor associated with a maximal clique is the product of all factors from the original distribution that were assigned to it. So for example, taking the figure 4.4(b) in the book, suppose that the Gibbs distribution was defined by the factor product

$$\phi(A, B) * \phi(A, D) * \phi(A, B, D) * \phi(B,C,D)$$

We can alternatively parametrize this over the maximal cliques $$\{A, B, D\}$$ and $$\{B, C, D\}$$, and the factor product then becomes

$$\psi(A, B, D) * \psi(B, C, D)$$

where

$$\psi(A, B, D) = \phi(A, B) * \phi(A, D) * \phi(A, B, D)$$ and $$\psi(B, C, D) = \phi(B, C, D)$$

Now you can see that these factorizations both define the same distribution over the graphical model, but it's not clear anymore from the second distribution what the first distribution is. This is what is meant by the bolded section - by using this parametrization you lose information because you assign multiple factors to a single "new" factor.