Pseudo-Marginal Metropolis–Hastings

Sampling When Likelihoods Are Intractable

In some models, the likelihood function cannot be evaluated directly. The pseudo-marginal Metropolis–Hastings algorithm addresses this by replacing exact likelihoods with unbiased estimates.

Key Idea

Instead of computing the true likelihood:

• an estimate is used

• the algorithm proceeds as if it were exact

Provided the estimate is unbiased, the method still converges to the correct distribution.

Why This Matters

This approach allows modelling in situations where:

• exact likelihoods are computationally infeasible

• simulations are required to approximate probabilities

Application at MorMag

Used in the Quant Lab for:

• complex models involving latent variables

• simulation-heavy frameworks

• high-dimensional inference problems

This enables modelling beyond analytically tractable systems.

Conclusion

Pseudo-marginal methods extend the reach of probabilistic modelling, allowing inference in complex environments where exact solutions are unavailable.