Time Series Sequential Monte Carlo

MCMC explores the full joint posterior over all time steps at once and is inherently offline — it requires all data to be available before sampling begins. SMC processes observations sequentially and produces a running approximation to the filtering posterior at each time step, making it suitable for online or real-time applications. The two can be combined in SMC samplers or particle MCMC algorithms.

The Kalman filter is the optimal and exact filter for linear-Gaussian state-space models and is far more efficient than the particle filter in that setting. Use the particle filter only when the transition or emission densities are nonlinear or the noise is non-Gaussian — exactly the cases where the Kalman filter is either an approximation (EKF, UKF) or inapplicable.

There is no universal answer. In low-dimensional states (1–5 dimensions), a few hundred to a few thousand particles often suffice. In higher-dimensional states, the required N can grow exponentially. Practical guidance is to monitor the effective sample size and run experiments with increasing N until estimates stabilise.

Filtering produces the posterior over the current hidden state given all observations up to and including the current time: p(x_t | y_{1:t}). Smoothing produces the posterior over a past state given all observations, including future ones: p(x_s | y_{1:T}) for s < T. Smoothing is retrospective and requires a backward pass; filtering is online and sequential.

Yes. Static parameters can be included in an augmented state and estimated jointly, though this often leads to particle impoverishment. More robust approaches include particle MCMC (PMCMC), SMC² (sequential SMC for both states and parameters), or adding small artificial noise to parameters and using the Liu-West filter.