Abstract The problem of estimation of the state of a linear dynamic system driven by white Gaussian noise with unknown covariance Q and observed by a linear function of the state contaminated by white Gaussian noise with unknown covariance R is considered. Bayesian recursion relations for the probability densities of the unknown random variables conditioned on all available data are developed. Initially a relation giving the a posteriori densities of Q and R conditioned on all available data is developed. However since the possible range of Q and R is generally unbounded and also because Q and R may have quite large dimension, a mechanization of the aforesaid algorithm by representing Q and R with a grid of possible values is considered unfeasible for any realistic problems. Therefore, the random variables Q and R are transformed to random variables representing the optimal gain and the covariance of the innovations process by use of the steady state Kalman filter relations. The resulting probability dens...