Bayesian analysis in the case of an estimated parameter following a stochastic process

We perform Bayesian analysis of the sequence of unknown means mi given observations Xi under the assumption that, for any k > 0, the first k members X1, X2, …, Xk are normally distributed with the mean (m1,…, mk ) and a known covariance matrix. It is assumed that the parameters m1,…, mk,… follow a Gaussian process We prove that, for any fixed k, the covariance matrices of marginal posterior distributions converge In the case of a Gaussian AR(1) process analytic expression for the asymptotic posterior structure is given