A sequential Monte Carlo method that represents a probability density by a set of weighted particles which are propagated, weighted by the likelihood of observations, and resampled to approximate posterior distributions in state-space models, especially when models are nonlinear or noise is non-Gaussian.

Principle

Approximate the filtering distribution by a discrete weighted empirical measure: propagate particles through the dynamics, assign importance weights according to the observation likelihood, and periodically resample to concentrate computational effort on high-weight regions while controlling weight degeneracy.

Demonstration

Localize a mobile robot with nonlinear range-bearing sensors: simulate many particles representing possible poses, predict each particle via motion model, weight particles by sensor likelihood, resample to focus on plausible poses, and use the empirical distribution for navigation decisions.

Misapplication

Using too few particles, neglecting resampling or resampling too frequently without jitter, which leads to sample impoverishment, weight collapse, or loss of diversity and consequently poor posterior approximation.

Consequence

Particle filters can approximate arbitrary posterior distributions online and handle severe nonlinearity and non-Gaussian noise, enabling sequential state estimation in contexts unsuitable for Kalman-type filters at the cost of computational load and stochastic error.

Reversal

Analytic Gaussian filters (Kalman and its variants) invert the approach by maintaining parametric Gaussian summaries rather than ensembles of samples, trading generality for closed-form efficiency when linear-Gaussian approximations are adequate.

Boundary

Effective for low-to-moderate state dimensions and when computational budget allows many particles; suffers from the curse of dimensionality as required particle counts grow exponentially with effective state dimension and from difficulty in handling highly informative observations.

Semantic Tension

Tension between particle filters and parametric filters (Kalman family) stems from the tradeoff between representational generality (particles approximate arbitrary shapes) and computational efficiency and analytic covariance propagation of Gaussian filters.

Synthesis

Particle filters realize Bayesian sequential estimation by maintaining a Monte Carlo approximation to the posterior: repeated propagate-weight-resample cycles transform ensembles into empirical approximations that capture nonlinearity and non-Gaussianity, while algorithmic choices (proposal, resampling, particle count) govern accuracy and computational cost.