In game theory, a Correlated Equilibrium (CE) is an equilibrium concept that generalizes the more well-known Nash Equilibrium. If the game is represented as a graphical game, the computational complexity of computing an optimum CE is exponential in the tree-width of the graph. In settings where this exact computation is not feasible, it is desirable to approximate the properties of the CE, such as its expected social utility and marginal probabilities. We study outer relaxations of this problem that yield approximate marginal strategies for the players under a variety of utility functions. Results on simulated games and in a real problem involving drug design indicate that our approximations can be highly accurate and can be successfully used when exact computation of CE is infeasible.