# Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC

arXiv: https://arxiv.org/abs/2104.03942 · HTML: https://mlsanity.com/paper/2104.03942/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 82 references that are in the map.
- First posted on arXiv 8 Apr 2021.

## Builds on (most cited first, 30 of 32)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models](https://mlsanity.com/paper/1501.03291/) (2015)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [On Markov chain Monte Carlo methods for tall data](https://mlsanity.com/paper/1505.02827/) (2015)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Patterns of Scalable Bayesian Inference](https://mlsanity.com/paper/1602.05221/) (2016)
- [Austerity in MCMC Land: Cutting the Metropolis-Hastings Budget](https://mlsanity.com/paper/1304.5299/) (2013)
- [Variational Bayesian Monte Carlo with Noisy Likelihoods](https://mlsanity.com/paper/2006.08655/) (2020)
- [Variational Bayesian Monte Carlo](https://mlsanity.com/paper/1810.05558/) (2018)
- [Generalized Posteriors in Approximate Bayesian Computation](https://mlsanity.com/paper/2011.08644/) (2020)
- [Parallel Gaussian process surrogate Bayesian inference with noisy likelihood evaluations](https://mlsanity.com/paper/1905.01252/) (2019)
- [GPS-ABC: Gaussian Process Surrogate Approximate Bayesian Computation](https://mlsanity.com/paper/1401.2838/) (2014)
- [Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations](https://mlsanity.com/paper/1402.1694/) (2014)
- [Efficient acquisition rules for model-based approximate Bayesian computation](https://mlsanity.com/paper/1704.00520/) (2017)
- [Asymptotically Optimal Exact Minibatch Metropolis-Hastings](https://mlsanity.com/paper/2006.11677/) (2020)
- [Improving Quadrature for Constrained Integrands](https://mlsanity.com/paper/1802.04782/) (2018)
- [Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels](https://mlsanity.com/paper/1403.5496/) (2014)
- [Evaluating Gaussian Process Metamodels and Sequential Designs for Noisy Level Set Estimation](https://mlsanity.com/paper/1807.06712/) (2018)
- [Adaptive, delayed-acceptance MCMC for targets with expensive likelihoods](https://mlsanity.com/paper/1509.00172/) (2015)
- [Adaptive Gaussian process approximation for Bayesian inference with expensive likelihood functions](https://mlsanity.com/paper/1703.09930/) (2017)
- [Accelerating ABC methods using Gaussian processes](https://mlsanity.com/paper/1401.1436/) (2014)
- [Robust Bayesian Synthetic Likelihood via a Semi-Parametric Approach](https://mlsanity.com/paper/1809.05800/) (2018)
- [Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases](https://mlsanity.com/paper/1506.05555/) (2015)
- [Batch simulations and uncertainty quantification in Gaussian process surrogate approximate Bayesian computation](https://mlsanity.com/paper/1910.06121/) (2019)
- [Query Efficient Posterior Estimation in Scientific Experiments via Bayesian Active Learning](https://mlsanity.com/paper/1702.01145/) (2017)

## Built on by (most cited first, 3 of 3)

- [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)
- [Measuring the right thing: justifying metrics in AI impact assessments](https://mlsanity.com/paper/2504.05007/) (2025)
- [A Critical Review of Monte Carlo Algorithms Balancing Performance and Probabilistic Accuracy with AI Augmented Framework](https://mlsanity.com/paper/2512.17968/) (2025)

## Abstract (arXiv)

We present a framework for approximate Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained due to computational constraints, which is becoming increasingly common for applications of complex models. We model the log-likelihood function using a Gaussian process (GP) and the main methodological innovation is to apply this model to emulate the progression that an exact Metropolis-Hastings (MH) sampler would take if it was applicable. Informative log-likelihood evaluation locations are selected using a sequential experimental design strategy until the MH accept/reject decision is done accurately enough according to the GP model. The resulting approximate sampler is conceptually simple and sample-efficient. It is also more robust to violations of GP modelling assumptions compared with earlier, related "Bayesian optimisation-like" methods tailored for Bayesian inference. We discuss some theoretical aspects and various interpretations of the resulting approximate MH sampler, and demonstrate its benefits in the context of Bayesian and generalised Bayesian likelihood-free inference for simulator-based statistical models.