The Discriminative Kalman Filter adopts the Kalman state model, p(zt|zt-1) = ηd(zt ; Azt-1, Γ), with initialization p(z0) = ηd(z0; 0, S) where S satisfies S=ASA'+Γ (so that the latent process is stationary) and uses the measurement model introduced above. Given these specifications, it follows that we may recursively approximate the posterior as Gaussian. Namely, if
p(zt-1|x1:t-1) ≈ ηd(zt-1 ; μt-1, Σt-1),then given a new observation Xt=xt, we have that
p(zt|x1:t ) ≈ ηd(zt ; μt , Σt )where
Mt = AΣt-1A' + Γ,
Σt = (Mt-1 + Q(xt )-1 - S-1)-1,
μt = Σt(Mt-1Aμt-1 + Q(xt )-1f(xt)).This approximation is functionally exact when Q(xt )-1 - S -1 is positive-definite; otherwise we let
Σt = (Mt-1 + Q(xt )-1)-1.In this way, the Discriminative Kalman Filter maintains fast, closed-form updates while allowing for a nonlinear relationship between the latent states and observations. When supervised training data is available, off-the-shelf nonlinear/nonparameteric regression tools can readily be used to learn the discriminatively specified observation model. In related work, we demonstrate how this framework can also be leveraged to ameliorate non-stationarities, or changes to the relationship between the latent states and observations, and increase the robustness of estimates.