The discriminative Kalman filter for nonlinear and non-Gaussian sequential Bayesian filtering

TL;DR

This paper introduces the discriminative Kalman filter (DKF), a new Bayesian filtering method that models the posterior directly, offering improved accuracy and ease of learning in high-dimensional observation scenarios, validated on synthetic and neural data.

Contribution

The paper proposes the DKF, a closed-form discriminative Bayesian filter that simplifies learning and enhances performance in nonlinear, high-dimensional observation models.

Findings

DKF outperforms standard Kalman filter in neural decoding tasks.

Highly nonlinear models for p(state|observation) are feasible with mild assumptions.

Substantial performance improvements demonstrated on synthetic and real neural data.

Abstract

The Kalman filter (KF) is used in a variety of applications for computing the posterior distribution of latent states in a state space model. The model requires a linear relationship between states and observations. Extensions to the Kalman filter have been proposed that incorporate linear approximations to nonlinear models, such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). However, we argue that in cases where the dimensionality of observed variables greatly exceeds the dimensionality of state variables, a model for $p(\text{state}|\text{observation})$ proves both easier to learn and more accurate for latent space estimation. We derive and validate what we call the discriminative Kalman filter (DKF): a closed-form discriminative version of Bayesian filtering that readily incorporates off-the-shelf discriminative learning techniques. Further, we demonstrate that given mild assumptions, highly non-linear models for $p(\text{state}|\text{observation})$ can be specified. We motivate and validate on synthetic datasets and in neural decoding from non-human primates, showing substantial increases in decoding performance versus the standard Kalman filter.