In this paper, the inverse Kalman filtering problem is addressed using a duality-based framework, where certain statistical properties of uncertainties in a dynamical model are recovered from observations of its posterior estimates. The duality relation in inverse filtering and inverse optimal control is established. It is shown that the inverse Kalman filtering problem can be solved using results from a well-posed inverse linear quadratic regulator. Identifiability of the considered inverse filtering model is proved and a unique covariance matrix is recovered by a least squares estimator, which is also shown to be statistically consistent. Effectiveness of the proposed methods is illustrated by numerical simulations.