# Adaptive, delayed-acceptance MCMC for targets with expensive likelihoods

arXiv: https://arxiv.org/abs/1509.00172 · HTML: https://mlsanity.com/paper/1509.00172/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2015.

## Built on by (most cited first, 9 of 9)

- [Accelerated Diffusion Models via Speculative Sampling](https://mlsanity.com/paper/2501.05370/) (2025)
- [A Survey of Monte Carlo Methods for Parameter Estimation](https://mlsanity.com/paper/2107.11820/) (2021)
- [A survey of Monte Carlo methods for noisy and costly densities with application to reinforcement learning and ABC](https://mlsanity.com/paper/2108.00490/) (2021)
- [SwISS: A Scalable Markov chain Monte Carlo Divide-and-Conquer Strategy](https://mlsanity.com/paper/2208.04080/) (2022)
- [Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC](https://mlsanity.com/paper/2104.03942/) (2021)
- [Bayesian score calibration for approximate models](https://mlsanity.com/paper/2211.05357/) (2022)
- [General bounds on the quality of Bayesian coresets](https://mlsanity.com/paper/2405.11780/) (2024)
- [Gaussian Processes to speed up MCMC with automatic exploratory-exploitation effect](https://mlsanity.com/paper/2109.13891/) (2021)
- [Multi-fidelity Hamiltonian Monte Carlo](https://mlsanity.com/paper/2405.05033/) (2024)

## Abstract (arXiv)

When conducting Bayesian inference, delayed acceptance (DA) Metropolis-Hastings (MH) algorithms and DA pseudo-marginal MH algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased estimate thereof, but a computationally cheap approximation is available. A first accept-reject stage is applied, with the cheap approximation substituted for the true posterior in the MH acceptance ratio. Only for those proposals which pass through the first stage is the computationally expensive true posterior (or unbiased estimate thereof) evaluated, with a second accept-reject stage ensuring that detailed balance is satisfied with respect to the intended true posterior. In some scenarios there is no obvious computationally cheap approximation. A weighted average of previous evaluations of the computationally expensive posterior provides a generic approximation to the posterior. If only the k-nearest neighbours have non-zero weights then evaluation of the approximate posterior can be made computationally cheap provided that the points at which the posterior has been evaluated are stored in a multi-dimensional binary tree, known as a KD-tree. The contents of the KD-tree are potentially updated after every computationally intensive evaluation. The resulting adaptive, delayed-acceptance [pseudo-marginal] Metropolis-Hastings algorithm is justified both theoretically and empirically. Guidance on tuning parameters is provided and the methodology is applied to a discretely observed Markov jump process characterising predator-prey interactions and an ODE system describing the dynamics of an autoregulatory gene network.