In this thesis, we study a special adaptive system with a nonlinear term. The model has an extreme value at an unknown point. Using Bayesian approach, we succeed in establishing a strong consistent control rule which makes the output as good as if the parameters are known. We also prove that in a simple linear regression model the posterior covariance matrix vanishes almost surely. And this result implies the strong consistency of Bayes estimates....[ Read more ]

In this thesis, we study a special adaptive system with a nonlinear term. The model has an extreme value at an unknown point. Using Bayesian approach, we succeed in establishing a strong consistent control rule which makes the output as good as if the parameters are known. We also prove that in a simple linear regression model the posterior covariance matrix vanishes almost surely. And this result implies the strong consistency of Bayes estimates.

In Chapter 1, we introduce martingale theories which are the important tools of this thesis. And Doob's martingale convergence results form the kernel part of chapter 1. Since the proofs of these theorems need monotone class theorem , concept of uniform integrability etc, we also record several theorems on these topics.

In Chapter 2, using Bayesian approach, we build up a control rule for a special model with an extreme value and prove the strong consistency.

In Chapter 3, we prove that the posterior matrix vanishes almost surely in a simple case of linear regression model. Hence we can get the strong consistency of Bayes estimates in this case.

In Chapter 4, we give conclusions and a short list of open problems.