Particle Filter

Definition

A particle filter (sequential Monte Carlo, SMC) is a simulation method for on-line filtering in a non-linear / non-Gaussian state-space model — recursively approximating the filtering distribution $f(\alpha_t\mid\mathcal F_t)$ of a Markovian latent state $\alpha_t$ given the data $\mathcal F_t=(y_1,\dots,y_t)$ observed so far. The continuous state is represented by a swarm of $M$ weighted particles $\{\alpha_t^j,\pi_t^j\}_{j=1}^M$ — a discrete distribution with random support that converges to the true filtering density as $M\to\infty$ . As each new observation arrives, the particles are propagated through the state transition, reweighted by the likelihood, and resampled, so the data can be processed in a single forward sweep. It is the Monte Carlo counterpart of the Kalman filter, which is exact only in the linear-Gaussian case.

Key Ideas

Random-support approximation. $f(\alpha_t\mid\mathcal F_t)\approx\sum_j \pi_t^j\,\delta_{\alpha_t^j}$ ; the "empirical filtering density" $\hat f(\alpha_{t+1}\mid\mathcal F_{t+1})\propto f(y_{t+1}\mid\alpha_{t+1})\sum_j\pi_t^j f(\alpha_{t+1}\mid\alpha_t^j)$ is sampled to produce the next generation of particles.
On-line, single-sweep. Unlike MCMC smoothers that reprocess the whole sample, a particle filter updates the state estimate recursively as data arrive — the natural tool for real-time/streaming state estimation.
Requirements. A known measurement density $f(y_t\mid\alpha_t)$ (evaluable up to proportionality) and the ability to simulate from the transition density $f(\alpha_{t+1}\mid\alpha_t)$ (and sometimes to evaluate it).
Weight degeneracy is the enemy. After a few steps the weights concentrate on a handful of particles (sample impoverishment); resampling combats this, and the design of the proposal determines how badly it bites.

How It Works

The bootstrap / SIR filter (Gordon–Salmond–Smith 1993). Treat $\tfrac1M\sum_j f(\alpha_{t+1}\mid\alpha_t^j)$ as a prior: (1) draw a parent $j$ with probability $\pi_t^j$ and propagate $\alpha_{t+1}\sim f(\cdot\mid\alpha_t^j)$ ; (2) reweight by the likelihood $f(y_{t+1}\mid\alpha_{t+1})$ ; (3) resample $M$ particles with those weights. Simple, but it propagates before seeing where the data are, so a surprising observation or a sharp likelihood leaves almost all particles with negligible weight.

The auxiliary particle filter (APF; Pitt–Shephard 1999). Augment the target with an auxiliary index $k$ for the parent particle and resample before propagating, using a look-ahead at the next observation: pre-weight each parent by the likelihood evaluated at a representative point of its predictive distribution (e.g. its mean $\mu_{t+1}^j$ ), select promising parents, propagate them, then apply a second-stage correction weight. This adapts the proposal to the incoming data and dramatically reduces degeneracy when observations are informative or extreme.

Full adaptation. A filter is fully adapted if it produces i.i.d. draws from the exact one-step target $f(\alpha_{t+1}\mid\mathcal F_{t+1})$ — the ideal proposal. The APF approximates full adaptation cheaply (and achieves it exactly in conditionally-Gaussian cases). One-step sampling can use sampling/importance-resampling, rejection sampling, or MCMC.

Why It Matters

Non-linear/non-Gaussian filtering. Particle filters give a general on-line solution where the Kalman filter cannot apply — target tracking, stochastic-volatility models, and other financial/engineering state-space problems.
The APF is the standard upgrade. Its look-ahead resampling is the default fix for the bootstrap filter's degeneracy and made particle filtering practical for informative-likelihood models.
Foundation for particle MCMC. Particle filters supply unbiased likelihood estimates that later underpin particle-MCMC and SMC² for full Bayesian parameter inference.

Open Questions