As I read it, Bishop is asserting that for each maximal clique $C$ we may define the potential as $\psi_C(x_C) = \prod_S U_S(x_S)$ where $S$ denotes cliques which are subsets of $C$ (and $U$ is the energy, as in the link from Li). This is how I interpret "[if $C$] is a maximum clique, and we define an arbitrary function over this clique, then including another factor defined over a subset of these variables would be redundant." It is potentially confusing because, as typical in statistics, the symbol $\psi$ is overloaded to have allow different functional form for any clique. In any case, Bishop is defining each maximal clique potential in terms of a factorization in terms of all subset cliques. Substitution gives back the other definition.

As I read it, Bishop is asserting that for each maximal clique $C$ we may define the potential as $\psi_C(x_C) = \prod_S U_S(x_S)$ where $S$ denotes cliques which are subsets of $C$ (and $U$ is the energy, as in the link from Li). This is how I interpret "[if $C$] is a maximum clique, and we define an arbitrary function over this clique, then including another factor defined over a subset of these variables would be redundant." It is potentially confusing because, as typical in statistics, the symbol $\psi$ is overloaded to allow different functional form for any clique. In any case, Bishop defines each maximal clique potential in terms of a factorization over all subset cliques. Substitution gives back the other definition.