Before revision:  In this paper, we use a generic model of a Bayesian game with the solution concept of Bayes Correlated Equilibria of Bergemann and Morris (2016) and show that if the information structure is rich enough and the equilibrium selection rule satisfies a weak invariance property, decomposition-based predictions are identical to equilibrium-based predictions.
After revision:  In this paper, we use a generic model of a Bayesian game with the solution concept of Bayes Correlated Equilibria of Bergemann and Morris (2016) and develop a method of counterfactual analysis that can be used for such situations. We show that, under an invariance condition for the equilibrium selection rule satisfies a weak invariance property, decomposition-based predictions are identical to equilibrium-based predictions.