Particle filter (sequential Monte Carlo)

A particle filter represents the posterior over the hidden state by a cloud of weighted samples (particles) rather than a Gaussian, so it can in principle track arbitrarily nonlinear dynamics and non-Gaussian, multimodal beliefs. Each step samples each particle forward through the dynamics, reweights it by how well it explains the new observation (the likelihood), and periodically resamples to concentrate particles where probability is high. As the number of particles grows the approximation converges to the true posterior.

The price is computation and a set of well-known failure modes. Naive sequential importance sampling suffers weight degeneracy, where one particle dominates; resampling fixes this but causes sample impoverishment, losing diversity, especially in high-dimensional state spaces where the number of particles needed can explode. In BCI the particle filter is valuable when the observation model is genuinely non-Gaussian — for example spike-based likelihoods or switching/multimodal intent — but for smooth Gaussian-ish problems a Kalman or unscented filter usually delivers equivalent accuracy far more cheaply, so particle filters are used selectively rather than by default.