On consistency of Bayes estimates in a certainty equivalence adaptive system

The authors analyze a control problem with data generated by the linear regression model where intercept and slope coefficients are unknown. They propose a certainty equivalence control rule based on Bayes estimates of the intercept and slope coefficients. It is shown that the control rule converges to the optimal control rule, which requires complete knowledge of intercept and slope coefficients. Furthermore, under the proposed control rule, if the total control cost tends to infinity, they show that the Bayes estimates for slope and intercept are consistent