# Variational Bayesian Monte Carlo

arXiv: https://arxiv.org/abs/1810.05558 · HTML: https://mlsanity.com/paper/1810.05558/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Oct 2018.

## Built on by (most cited first, 20 of 20)

- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
- [Parallel MCMC Without Embarrassing Failures](https://mlsanity.com/paper/2202.11154/) (2022)
- [Conditional Bayesian Quadrature](https://mlsanity.com/paper/2406.16530/) (2024)
- [PyVBMC: Efficient Bayesian inference in Python](https://mlsanity.com/paper/2303.09519/) (2023)
- [Fast and robust Bayesian Inference using Gaussian Processes with GPry](https://mlsanity.com/paper/2211.02045/) (2022)
- [High Frequency Matters: Uncertainty Guided Image Compression with Wavelet Diffusion](https://mlsanity.com/paper/2407.12538/) (2024)
- [Efficient Autoregressive Inference for Transformer Probabilistic Models](https://mlsanity.com/paper/2510.09477/) (2025)
- [Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC](https://mlsanity.com/paper/2104.03942/) (2021)
- [Interpolating between sampling and variational inference with infinite stochastic mixtures](https://mlsanity.com/paper/2110.09618/) (2021)
- [Compound Density Networks for Risk Prediction using Electronic Health Records](https://mlsanity.com/paper/2208.01320/) (2022)
- [PyBADS: Fast and robust black-box optimization in Python](https://mlsanity.com/paper/2306.15576/) (2023)
- [PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inference](https://mlsanity.com/paper/2510.13763/) (2025)
- [Cyclical Variational Bayes Monte Carlo for Efficient Multi-Modal Posterior Distributions Evaluation](https://mlsanity.com/paper/2202.11645/) (2022)
- [Active Nearest Neighbor Regression Through Delaunay Refinement](https://mlsanity.com/paper/2206.08061/) (2022)
- [Bayesian Adaptive Calibration and Optimal Design](https://mlsanity.com/paper/2405.14440/) (2024)
- [Normalizing Flow Regression for Bayesian Inference with Offline Likelihood Evaluations](https://mlsanity.com/paper/2504.11554/) (2025)
- [Parallelized Acquisition for Active Learning using Monte Carlo Sampling](https://mlsanity.com/paper/2305.19267/) (2023)
- [Measuring the right thing: justifying metrics in AI impact assessments](https://mlsanity.com/paper/2504.05007/) (2025)
- [Digital twin for virtual sensing of ferry quays via a Gaussian Process Latent Force Model](https://mlsanity.com/paper/2506.14925/) (2025)
- [Monte Carlo Stochastic Depth for Uncertainty Estimation in Deep Learning](https://mlsanity.com/paper/2604.12719/) (2026)

## Abstract (arXiv)

Many probabilistic models of interest in scientific computing and machine learning have expensive, black-box likelihoods that prevent the application of standard techniques for Bayesian inference, such as MCMC, which would require access to the gradient or a large number of likelihood evaluations. We introduce here a novel sample-efficient inference framework, Variational Bayesian Monte Carlo (VBMC). VBMC combines variational inference with Gaussian-process based, active-sampling Bayesian quadrature, using the latter to efficiently approximate the intractable integral in the variational objective. Our method produces both a nonparametric approximation of the posterior distribution and an approximate lower bound of the model evidence, useful for model selection. We demonstrate VBMC both on several synthetic likelihoods and on a neuronal model with data from real neurons. Across all tested problems and dimensions (up to D = 10), VBMC performs consistently well in reconstructing the posterior and the model evidence with a limited budget of likelihood evaluations, unlike other methods that work only in very low dimensions. Our framework shows great promise as a novel tool for posterior and model inference with expensive, black-box likelihoods.