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Particle Filter

A sequential Monte Carlo method that approximates a hidden state's posterior distribution using a large collection of weighted random samples ('particles'), designed for state-space models where dynamics or noise are nonlinear or non-Gaussian and the Kalman filter's exact formulas no longer apply.

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In Plain English

The Kalman filter's clean predict-update formulas only work because linear transformations of Gaussian distributions stay Gaussian, do exactly the same math on a nonlinear system or a lumpy, non-Gaussian noise distribution and the formulas break down. The particle filter sidesteps this entirely by representing the hidden state's uncertainty not as a formula but as a swarm of many individual guesses, particles, each carrying a weight reflecting how consistent it is with the data so far. As new observations arrive, particles that no longer fit well get down-weighted (and eventually discarded and replaced, resampled), and the swarm as a whole tracks the true, possibly very irregularly-shaped posterior distribution, no Gaussian assumption required.

Definition

A particle filter (sequential Monte Carlo) approximates the filtering distribution $P (X_{t} ∣ Y_{1 : t})$ for a general (possibly nonlinear, non-Gaussian) state-space model using a set of $N$ weighted particles ${X_{t}^{(i)}, w_{t}^{(i)}}_{i = 1}^{N}$ . At each time step, particles are propagated forward through the (possibly nonlinear) state equation, then reweighted according to how well each particle's predicted observation matches the actual new observation $Y_{t}$ , via the likelihood $P (Y_{t} ∣ X_{t}^{(i)})$ . To prevent weight degeneracy (most weight collapsing onto very few particles after many iterations), a resampling step periodically draws a new particle set with replacement, proportional to current weights, refocusing computational effort on the most probable regions of the state space. As $N \to \infty$ , the weighted particle empirical distribution converges to the true filtering distribution, giving particle filters an asymptotic correctness guarantee that holds for essentially arbitrary nonlinear, non-Gaussian state-space models, unlike the Kalman filter's linear-Gaussian restriction.

Properties

Resampling is the specific technical fix that makes particle filters practical rather than a theoretical curiosity, without it, after enough time steps nearly all particles end up with negligible weight (weight degeneracy), wasting computation on particles that no longer represent plausible states.
Particle filters trade the Kalman filter's exact, closed-form optimality for broad applicability, they handle essentially any nonlinear dynamics or non-Gaussian noise (multimodal posteriors, heavy tails) that the Kalman filter's Gaussian machinery simply cannot represent, at the cost of being a Monte Carlo approximation rather than an exact solution.
Applications span robotics (simultaneous localization and mapping, tracking a robot's position from noisy, nonlinear sensor readings), object tracking in computer vision, and nonlinear financial models like stochastic volatility, anywhere the linear-Gaussian Kalman filter assumptions are simply too restrictive to apply.

At a Glance

Applies togeneral (nonlinear, non-Gaussian) state-space models

Representationthe posterior approximated by a weighted set of

N

random particles

Update mechanismpropagate, reweight by observation likelihood, then resample to prevent weight collapse

Guaranteeconverges to the true filtering distribution as

N \to \infty

Last updated September 8, 2026← Back to the Dictionary