What is Sequential Monte Carlo?
Sequential Monte Carlo is a family of algorithms that approximates a sequence of probability distributions with weighted particles updated through propagation, reweighting, resampling, and optional rejuvenation.
Quick Facts
| Specification | Official Specification |
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How It Works
Move a population through target distributions
Let pi_0, ..., pi_T be target distributions, with pi_0 easy to sample and pi_T the desired target. Each stage propagates particles with a forward kernel and updates an incremental importance weight that corrects the difference between consecutive targets and proposals. A backward kernel can make this correction tractable even when the propagated marginal density is unavailable.
The SMC sampler tutorial by Dai and colleagues distinguishes general fixed-space SMC samplers from Particle Filters, where targets are filtering distributions indexed by observation time.
Resample selectively and rejuvenate diversity
Normalized weights define a particle Effective Sample Size such as 1 / sum(w_i^2). When it falls below a declared threshold, resampling removes low-weight particles and duplicates high-weight ones. This controls weight degeneracy but creates shared ancestry and does not generate new information.
Mutation kernels, often MCMC transitions invariant to the current target, restore diversity after resampling. The number of mutation steps, resampling method, threshold, and target schedule are part of the estimator and must be recorded.
Control path, genealogical, and Monte Carlo error
Large gaps between consecutive targets can collapse weights before mutation has a chance to help. Excessive resampling can impoverish ancestry, while too little resampling leaves most computation on negligible particles. High dimension can make either failure abrupt.
Track weight ESS before resampling, unique ancestors, resampling events, mutation acceptance and movement, normalizing-constant increments, and replicated-system variability. Particle weight ESS is not the autocorrelation-based MCMC ESS; both may be relevant when SMC uses MCMC rejuvenation.
Key Characteristics
- Represents each target with a weighted particle population
- Moves through observations or artificial bridging distributions
- Uses incremental importance weights between stages
- Resamples particles when weights become too concentrated
- Can apply MCMC mutation to restore particle diversity
- Can estimate ratios of normalizing constants as a by-product
Common Use Cases
- Online inference in nonlinear non-Gaussian state-space models
- Tempered sampling from a prior to a difficult posterior
- Sequential Bayesian updating as observations arrive
- Estimating marginal likelihoods and normalizing constants
- Rare-event and constrained-distribution simulation
Example
Loading code...Frequently Asked Questions
How does Sequential Monte Carlo differ from MCMC?
MCMC usually evolves one or more dependent chains for a fixed target. SMC evolves a weighted particle population through a sequence of targets, adds resampling interaction, and can estimate normalizing-constant ratios.
Is every Sequential Monte Carlo method a Particle Filter?
No. Particle Filters are SMC methods whose target sequence usually follows filtering distributions as observations arrive. SMC samplers can instead traverse fixed-dimensional tempered, partial-posterior, or constrained targets unrelated to real time.
Why does SMC resample particles?
Repeated importance weighting can concentrate nearly all mass on a few particles. Resampling reallocates computation toward those particles, but duplicated ancestry reduces diversity, so mutation and genealogical diagnostics remain necessary.
Is particle ESS the same as MCMC ESS?
No. Particle ESS summarizes concentration of normalized importance weights at one stage. MCMC ESS summarizes information loss from serial dependence for an estimand. An SMC algorithm with MCMC mutation may need both diagnostics.
What makes an SMC target schedule fail?
Consecutive targets may overlap too little, causing abrupt weight collapse and loss of modes. Monitor pre-resampling ESS, unique ancestors, incremental normalizing constants, mutation movement, and replicated-system variability, then refine the path or increase resources.