# Variational Bayes with Synthetic Likelihood

arXiv: https://arxiv.org/abs/1608.03069 · HTML: https://mlsanity.com/paper/1608.03069/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Aug 2016.

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

- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Variational methods for simulation-based inference](https://mlsanity.com/paper/2203.04176/) (2022)
- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Simulation-based Bayesian inference under model misspecification](https://mlsanity.com/paper/2503.12315/) (2025)
- [Quasi Black-Box Variational Inference with Natural Gradients for Bayesian Learning](https://mlsanity.com/paper/2205.11568/) (2022)
- [Wasserstein Gaussianization and Efficient Variational Bayes for Robust Bayesian Synthetic Likelihood](https://mlsanity.com/paper/2305.14746/) (2023)

## Abstract (arXiv)

Synthetic likelihood is an attractive approach to likelihood-free inference when an approximately Gaussian summary statistic for the data, informative for inference about the parameters, is available. The synthetic likelihood method derives an approximate likelihood function from a plug-in normal density estimate for the summary statistic, with plug-in mean and covariance matrix obtained by Monte Carlo simulation from the model. In this article, we develop alternatives to Markov chain Monte Carlo implementations of Bayesian synthetic likelihoods with reduced computational overheads. Our approach uses stochastic gradient variational inference methods for posterior approximation in the synthetic likelihood context, employing unbiased estimates of the log likelihood. We compare the new method with a related likelihood free variational inference technique in the literature, while at the same time improving the implementation of that approach in a number of ways. These new algorithms are feasible to implement in situations which are challenging for conventional approximate Bayesian computation (ABC) methods, in terms of the dimensionality of the parameter and summary statistic.