In a previous paper the convenience of using Martingale theory in the analysis of Bayesian least-squares estimation was demonstrated. However, certain restrictions had to be imposed on either the feedback structure or on the initial values for the estimation. In the present paper these restrictions are removed, and necessary and sufficient conditions for strong consistency (in a Bayesian sense) are given for the Gaussian white noise case without any assumptions on closed loop stability or on the feedback structure. In the open-loop case the poles are shown to be consistently estimated, almost everywhere, and in the closed-loop case certain choices of control law are shown to assure consistency. Finally adaptive control laws are treated, and implicit self-tuning regulators are shown to converge to the desired control laws.