A Bayesian solution to the problem of state estimation in an unknown noise environment†

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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 density is reduced to the posterioridensity of the optimal gain by the law of total probability. Based on the recursion relations for this density a practical, new, recursive algorithm for optimal state estimation in the presence of noise with unknown noise covariance is developed. Two examples are presented showing the results of the new algorithm.