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

Area of Science:

• State-space modeling
• Machine learning
• Signal processing

Background:

• The Kalman filter is optimal for linear Gaussian state-space models.
• Extended and unscented Kalman filters handle nonlinearities via linearization.
• Existing methods struggle with highly nonlinear or non-Gaussian observation models.

Discussion:

• The discriminative Kalman filter (DKF) approximates the posterior distribution $p(\text{state}|\text{observation})$ as Gaussian.
• This approach is more accurate and easier to learn than modeling $p(\text{observation}|\text{state})$ for nonlinear/non-Gaussian models.
• DKF performance improves with higher-dimensional observations, drawing from the Bernstein-von Mises theorem.

Key Insights:

• DKF offers computational efficiency similar to the Kalman filter.
• It outperforms standard Kalman extensions in nonlinear/non-Gaussian scenarios.
• DKF integrates seamlessly with nonlinear regression techniques for learning observation models.

Outlook:

• DKF can be faster than particle filters with comparable precision.
• Potential for broader applications in real-time control systems.
• Further extensions and theoretical investigations are warranted.