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Given a stationary state-space model that relates a sequence of hidden statesand corresponding measurements or observations, Bayesian filtering provides aprincipled statistical framework for inferring the posterior distribution ofthe current state given all measurements up to the present time. For example,the Apollo lunar module implemented a Kalman filter to infer its location froma sequence of earth-based radar measurements and land safely on the moon.  To perform Bayesian filtering, we require a measurement model that describesthe conditional distribution of each observation given state. The Kalman filtertakes this measurement model to be linear, Gaussian. Here we show how anonlinear, Gaussian approximation to the distribution of state givenobservation can be used in conjunction with Bayes' rule to build a nonlinear,non-Gaussian measurement model. The resulting approach, called theDiscriminative Kalman Filter (DKF), retains fast closed-form updates for theposterior. We argue there are many cases where the distribution of state givenmeasurement is better-approximated as Gaussian, especially when thedimensionality of measurements far exceeds that of states and the Bernstein-vonMises theorem applies. Online neural decoding for brain-computer interfacesprovides a motivating example, where filtering incorporates increasinglydetailed measurements of neural activity to provide users control over externaldevices. Within the BrainGate2 clinical trial, the DKF successfully enabledthree volunteers with quadriplegia to control an on-screen cursor in real-timeusing mental imagery alone. Participant "T9" used the DKF to type out messageson a tablet PC.

A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding