What is Particle Markov Chain Monte Carlo?
Particle Markov Chain Monte Carlo is a family of methods that embeds particle filtering or conditional Sequential Monte Carlo inside MCMC to target parameter and latent-state posteriors in state-space models.
Quick Facts
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How It Works
Target an extended distribution, not a plug-in posterior
PMMH augments the state with all random variables used by a Particle Filter. Replacing an intractable likelihood by a nonnegative unbiased estimate inside the Metropolis-Hastings ratio then leaves an extended target invariant whose parameter marginal is the desired posterior. The estimator may be noisy, but it cannot be an arbitrary biased score.
Andrieu, Doucet, and Holenstein's original PMCMC paper proves the extended-space construction and introduces several particle MCMC kernels. The exactness claim depends on its assumptions, not merely on running particles inside an MCMC loop.
Retain the current likelihood estimate after rejection
A PMMH state includes both the parameter and its current likelihood estimate. Propose a parameter, run a fresh particle likelihood estimator for that proposal, and accept using prior, proposal, and estimated-likelihood terms. If rejected, retain both the current parameter and its existing estimate. Recomputing the current estimate every iteration is Monte Carlo within Metropolis and generally targets a different distribution.
Efficiency is sensitive to the variance and correlation of estimated log likelihoods. More particles reduce noise but increase cost; better proposals, correlated pseudo-marginal updates, and blocking can improve movement without changing the intended posterior when derived correctly.
Distinguish PMMH from Particle Gibbs
Particle Gibbs updates a latent trajectory with conditional SMC that preserves one reference path, then updates parameters conditionally. Basic conditional SMC can retain most of a long reference path because particle ancestries collapse; ancestor sampling, backward simulation, or blocking can improve path movement.
Report the PMCMC variant, particle count, resampling rule, likelihood-estimator variance, parameter acceptance, bulk and tail Effective Sample Size per unit time, trajectory refresh statistics, multiple-chain agreement, and sensitivity to particle count. Validate small instances against exact enumeration or a Kalman calculation where possible.
Key Characteristics
- Combines an outer MCMC kernel with particle-based latent-state computation
- Includes PMMH, Particle Gibbs, and particle independent MH variants
- Can preserve exact target marginals under an extended-space construction
- Requires a nonnegative unbiased likelihood estimator for standard PMMH
- Trades particle count against likelihood noise and runtime
- Can mix poorly through sticky estimates or collapsed latent ancestries
Common Use Cases
- Joint parameter and latent-state inference in state-space models
- Bayesian inference for nonlinear or non-Gaussian time series
- Parameter learning in stochastic volatility models
- Latent epidemic, ecological, or tracking models
- Posterior inference when a Particle Filter estimates likelihoods
Example
Loading code...Frequently Asked Questions
Is Particle MCMC an approximate inference method?
Finite particles approximate internal computations, but standard PMCMC kernels can retain the exact desired parameter or trajectory marginal on an extended space. That statement requires the specific unbiased-estimator or conditional-SMC assumptions to hold.
What is the difference between PMMH and Particle Gibbs?
PMMH proposes parameters and uses a particle likelihood estimate in an MH ratio. Particle Gibbs uses conditional SMC to update a latent trajectory while preserving a reference path, usually alternating with conditional parameter updates.
Why must PMMH retain the current likelihood estimate after rejection?
The random estimate is part of the extended Markov state. Refreshing it after rejecting the proposal changes the transition kernel and usually loses the pseudo-marginal invariance argument.
How many particles should Particle MCMC use?
Choose particles by measuring estimator variance, chain mixing, and effective samples per unit time. Too few can create sticky chains; too many waste computation. The useful count grows with sequence length and model difficulty.
How can Particle MCMC be validated?
Use exact enumeration, Kalman methods, or dense numerical integration on a small shared model; repeat chains and particle counts; and compare posterior summaries, likelihood-estimator behavior, trajectory movement, and runtime.