Sequential Monte Carlo

MCMC generates a Markov chain that converges to a fixed target posterior; it is designed for static problems where all data are available at once. SMC propagates a particle swarm in parallel and is designed for sequential or online settings where data arrive over time. SMC also provides an unbiased estimate of the marginal likelihood, which MCMC does not. SMC samplers can be used for static problems as well, often with better multimodal coverage than MCMC.

There is no universal rule. As a starting point, use N = 1000 and check the ESS over time. If ESS falls chronically below N/2, double N. For high-dimensional states, requirements grow steeply and alternative approaches (Rao-Blackwellisation, dimension reduction) should be considered alongside increasing N.

Without resampling, importance weights degenerate over time until a single particle carries all the weight and the approximation collapses. Resampling kills low-weight particles and replicates high-weight ones, maintaining representational diversity. The drawback is that resampling introduces Monte Carlo variance and can cause particle impoverishment. Triggering resampling only when ESS drops below a threshold (adaptive resampling) is the standard mitigation.

Use the Kalman filter when your model is linear and all noise is Gaussian — it is exact and computationally cheap. When the dynamics are nonlinear or the noise is non-Gaussian, the Kalman filter requires linearisation (EKF) or sigma-point approximation (UKF), both of which can be inaccurate. SMC makes no such approximation and will be more accurate at the cost of higher computation.

Particle filter is the most common name for SMC applied to state-space models where observations arrive sequentially and the goal is to track a hidden state. The terms 'particle filter' and 'SMC' are often used interchangeably; 'SMC sampler' specifically refers to the offline variant for static posterior computation introduced by Del Moral et al. (2006).