What is Pseudo-Marginal MCMC?
Pseudo-Marginal MCMC is a Markov Chain Monte Carlo approach that replaces an unavailable likelihood with a nonnegative unbiased random estimate while preserving the desired parameter posterior as a marginal distribution.
Quick Facts
| Specification | Official Specification |
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How It Works
Require an unbiased likelihood, not an unbiased log likelihood
For parameter theta, suppose random variables u produce L_hat(theta, u) >= 0 with E[L_hat(theta, u)] = L(theta). The augmented density proportional to prior(theta) L_hat(theta, u) m(u | theta) integrates to the desired posterior in theta. A log-likelihood estimate with zero mean error does not imply that its exponent is unbiased because Jensen's inequality intervenes.
Andrieu and Roberts' foundational paper formalizes this pseudo-marginal construction. Nonnegativity, unbiasedness, and the estimator's sampling law are structural assumptions.
Carry estimator randomness inside the Markov state
At each iteration, propose theta', draw new estimator randomness u', and use prior(theta') L_hat(theta', u') plus proposal terms in the acceptance ratio. If the proposal is rejected, retain both theta and the current u, or equivalently its current likelihood estimate. Refreshing the denominator estimate while keeping the parameter produces Monte Carlo within Metropolis, which is generally biased.
Particle Marginal Metropolis-Hastings is a prominent special case in which a Particle Filter supplies an unbiased likelihood estimate. Importance samplers, latent-variable estimators, and randomized truncations can also qualify when their mathematical conditions hold.
Control estimator noise rather than chasing acceptance alone
Occasionally large likelihood estimates can trap a chain because ordinary proposals struggle to exceed the current augmented-state value. Increasing estimator effort reduces noise but costs more per iteration. Correlating current and proposed estimator randomness can reduce variance of the log acceptance ratio when the resulting kernel preserves the correct auxiliary distribution.
Report estimator construction, computational effort, variance and tails of estimated Log Likelihood at representative parameters, acceptance, run lengths of repeated states, bulk and tail Effective Sample Size per unit time, multiple-chain agreement, and sensitivity to estimator effort. Validate on a case with an exactly available likelihood before relying on the intractable model.
Key Characteristics
- Targets a parameter posterior through an augmented estimator state
- Requires a nonnegative unbiased estimate of the likelihood
- Uses the random likelihood estimate in a Metropolis-Hastings ratio
- Retains the current estimate whenever a proposal is rejected
- Includes particle marginal methods as important special cases
- Can become sticky when likelihood-estimator noise is large
Common Use Cases
- Bayesian inference with latent variables that can be importance sampled
- Parameter inference using Particle Filter likelihood estimates
- Doubly intractable or simulator-based model research
- Exact-approximate alternatives to deterministic likelihood surrogates
- Benchmarking correlated likelihood-estimator designs
Example
Loading code...Frequently Asked Questions
Why is Pseudo-Marginal MCMC called exact?
Under its assumptions, the augmented chain has the desired parameter posterior as an exact marginal despite using random likelihood estimates. Finite chains still have Monte Carlo error and may mix too slowly for a useful estimate.
Can Pseudo-Marginal MCMC use an unbiased log-likelihood estimate?
Not by that fact alone. Standard pseudo-marginal theory requires the likelihood estimate itself to be nonnegative and unbiased. Exponentiating an unbiased log estimate generally creates bias.
What makes a pseudo-marginal chain sticky?
A rare overestimated likelihood can become the current augmented state, causing many ordinary proposals to be rejected. Heavy estimator tails or high log-likelihood variance therefore create long repeated-state runs.
How does Pseudo-Marginal MCMC differ from noisy MCMC?
Pseudo-marginal MCMC retains the current estimator randomness after rejection and has an extended-space invariant target. A noisy or Monte Carlo-within-Metropolis method refreshes estimates differently and generally introduces approximation bias.
How should a Pseudo-Marginal MCMC implementation be validated?
First compare it with ordinary MH on a model whose likelihood is exactly available. Then vary estimator effort and random seeds while checking posterior summaries, repeated-state runs, effective sample size per time, and estimator-tail behavior.