This paper presents a two-stage estimator for bias and state filtering in discrete-time stochastic linear systems affected by unknown inputs or disturbances. We show that the state estimate can be expressed as X-k/k = (X) over tilde(k/k) + beta(k/k)b(k/k) where (X) over tilde(k/k) is a bias-free state estimate and b(k/k) the optimal estimate of constant bias. The proposed two-stage estimator is based on an alternate derivation of the unbiased minimum variance estimator with unknown exogenous inputs developed by Darouach and Zasadzinski (1997, Automatica 33. 717-719).