This paper studies a new refinement for Perfect Bayesian equilibrium, Bayesian equilibrium by iterative conjectures (BEIC). BEIC requires players to make predictions, starting from first order uninformative predictive distribution functions (or conjectures) and keep updating with statistical decision theoretic and game theoretic reasoning until a convergence of conjectures is achieved. In a BEIC, conjectures are consistent with the equilibrium (or equilibriums) they supported and so rationality is achieved for actions, strategies and conjectures. The paper compares the solutions achieved under the BEIC approach and the perfect Bayesian equilibrium approach (including its two popular refinements, sequential equilibrium and the intuitive criterion). The BEIC approach has at least three advantages over the Perfect Bayesian Equilibrium approach: it normally generates only a compelling and intuitive unique equilibrium; it does not requires the specification of off-equilibrium beliefs which could be quite arbitrary and; it resolves inconsistencies in equilibrium results between sub-game Perfect Equilibrium (by backward induction) and Perfect Bayesian Equilibrium (and sequential equilibrium and intuitive criterion).