p(z_{t-1}|x_{1:t-1}) ≈ η_{d}(z_{t-1} ; μ_{t-1}, Σ_{t-1}),

then given a new observation Xp(z_{t}|x_{1:t} ) ≈ η_{d}(z_{t} ; μ_{t} , Σ_{t} )

whereM_{t} = AΣ_{t-1}A' + Γ,

Σ_{t} = (M_{t}^{-1} + Q(x_{t} )^{-1} - S^{-1})^{-1},

μ_{t} = Σ_{t}(M_{t}^{-1}Aμ_{t-1} + Q(x_{t} )^{-1}f(x_{t})).

This approximation is functionally exact when Q(xΣ_{t} = (M_{t}^{-1} + Q(x_{t} )^{-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.- M. Burkhart.
*Discriminative Bayesian filtering lends momentum to the stochastic Newton method for minimizing log-convex functions.*Optimization Letters 17 (2023) [pdf] [MR4557438] - M. Burkhart, D. Brandman, B. Franco, L. Hochberg, & M. Harrison.
*The Discriminative Kalman Filter for Bayesian Filtering with Nonlinear and Nongaussian Observation Models.*Neural Computation 32 (2020) [pdf] [MR4101168] - M. Burkhart.“A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding.” Ph.D. Dissertation, Brown University (2019) [pdf] [MR4158190]
- D. Brandman, M. Burkhart, J. Kelemen, B. Franco, M. Harrison, & L. Hochberg.
*Robust Closed-Loop Control of a Cursor in a Person with Tetraplegia using Gaussian Process Regression.*Neural Computation 30 (2018) [pdf] [MR3873814] - D. Brandman, T. Hosman, J. Saab, M. Burkhart, B. Shanahan, J. Ciancibello, et al.
*Rapid calibration of an intracortical brain computer interface for people with tetraplegia.*Journal of Neural Engineering 15 (2018)