Minimum bias priors for estimating additive terms in state-space models

This paper considers estimation of parametrized additive terms (also sometimes called "bias terms") in linear state space models. Estimation of the state as well as the random parameters, which may have an arbitrary prior and which may appear in nonlinear functions, is done in a Bayesian framework. It is shown how the complete posterior density function may be recursively and exactly evaluated. Closed form expressions for both the deterministic and stochastic Cramer-Rao bounds are derived. The asymptotic behavior of the Bayesian minimum mean-square-error estimator as a function of the prior density is then examined. An "adaptive prior" is introduced and shown to improve the performance of the estimator within a realization. The proposed adaptive prior yields an estimate whose expected value tends most quickly to the true parameter, i.e. has minimum bias.