The Discriminative Kalman Filter for Bayesian Filtering with Nonlinear and Nongaussian Observation Models

The Kalman filter provides a simple and efficient algorithm to compute the posterior distribution for state-space models where both the latent state and measurement models are linear and gaussian. Extensions to the Kalman filter, including the extended and unscented Kalman filters, incorporate linearizations for models where the observation model $p\left(\text{observation}\right|\text{state})$ is nonlinear. We argue that in many cases, a model for $p\left(\text{state}\right|\text{observation})$ proves both easier to learn and more accurate for latent state estimation. Approximating $p\left(\text{state}\right|\text{observation})$ as gaussian leads to a new filtering algorithm, the discriminative Kalman filter (DKF), which can perform well even when $p\left(\text{observation}\right|\text{state})$ is highly nonlinear and/or nongaussian. The approximation, motivated by the Bernstein-von Mises theorem, improves as the dimensionality of the observations increases. The DKF has computational complexity similar to the Kalman filter, allowing it in some cases to perform much faster than particle filters with similar precision, while better accounting for nonlinear and nongaussian observation models than Kalman-based extensions. When the observation model must be learned from training data prior to filtering, off-the-shelf nonlinear and nonparametric regression techniques can provide a gaussian model for $p\left(\text{observation}\right|\text{state})$ that cleanly integrates with the DKF. As part of the BrainGate2 clinical trial, we successfully implemented gaussian process regression with the DKF framework in a brain-computer interface to provide real-time, closed-loop cursor control to a person with a complete spinal cord injury. In this letter, we explore the theory underlying the DKF, exhibit some illustrative examples, and outline potential extensions.