# Correctness of sequential Monte Carlo

How does sequential Monte Carlo work?

Sequential Monte Carlo (SMC) internally utilizes Importance sampling: we approximate a series of target distributions ${P(\vec{x}_t)}$ using a weighted particle collection. Expectations with respect to any intermediate target can be computed by using the particle weights.

The key SMC ingredient is an update step which allows the introduction of new variables (possibly including new constrained observations) for $P(\vec{x}_{t+1})$. The update step evolves the particle population forward, and incrementally updates the existing importance weight vector $\vec{w}$ so that the new updated population is properly weighted (in the sense of importance sampling) with respect to the new target.

Pierre Del Moral, Arnaud Doucet, Ajay Jasra 2006 provides a general formulation of sequential Monte Carlo inference - including a description of the usage of Markov chain Monte Carlo kernels to evolve the particle population for a fixed target in between SMC update moves. This variant of SMC is often called **rejuvenation move SMC**.