# Pseudo-Marginal Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/1607.02516 · HTML: https://mlsanity.com/paper/1607.02516/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2016.

## Built on by (most cited first, 6 of 6)

- [Bayesian model calibration for block copolymer self-assembly: Likelihood-free inference and expected information gain computation via measure transport](https://mlsanity.com/paper/2206.11343/) (2022)
- [Revisiting the Effects of Stochasticity for Hamiltonian Samplers](https://mlsanity.com/paper/2106.16200/) (2021)
- [Repelling-Attracting Hamiltonian Monte Carlo](https://mlsanity.com/paper/2403.04607/) (2024)
- [Neural Bayes estimation and selection for complex bivariate extremal dependence models](https://mlsanity.com/paper/2503.23156/) (2025)
- [Tempered Sequential Monte Carlo for Trajectory and Policy Optimization with Differentiable Dynamics](https://mlsanity.com/paper/2604.21456/) (2026)
- [Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models](https://mlsanity.com/paper/2605.20345/) (2026)

## Abstract (arXiv)

Bayesian inference in the presence of an intractable likelihood function is computationally challenging. When following a Markov chain Monte Carlo (MCMC) approach to approximate the posterior distribution in this context, one typically either uses MCMC schemes which target the joint posterior of the parameters and some auxiliary latent variables, or pseudo-marginal Metropolis--Hastings (MH) schemes. The latter mimic a MH algorithm targeting the marginal posterior of the parameters by approximating unbiasedly the intractable likelihood. However, in scenarios where the parameters and auxiliary variables are strongly correlated under the posterior and/or this posterior is multimodal, Gibbs sampling or Hamiltonian Monte Carlo (HMC) will perform poorly and the pseudo-marginal MH algorithm, as any other MH scheme, will be inefficient for high dimensional parameters. We propose here an original MCMC algorithm, termed pseudo-marginal HMC, which combines the advantages of both HMC and pseudo-marginal schemes. Specifically, the pseudo-marginal HMC method is controlled by a precision parameter N, controlling the approximation of the likelihood and, for any N, it samples the marginal posterior of the parameters. Additionally, as N tends to infinity, its sample trajectories and acceptance probability converge to those of an ideal, but intractable, HMC algorithm which would have access to the marginal posterior of parameters and its gradient. We demonstrate through experiments that pseudo-marginal HMC can outperform significantly both standard HMC and pseudo-marginal MH schemes.