# Divide and Couple: Using Monte Carlo Variational Objectives for Posterior Approximation

arXiv: https://arxiv.org/abs/1906.10115 · HTML: https://mlsanity.com/paper/1906.10115/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2019.

## Built on by (most cited first, 14 of 14)

- [MCMC Variational Inference via Uncorrected Hamiltonian Annealing](https://mlsanity.com/paper/2107.04150/) (2021)
- [Black Box Variational Inference with a Deterministic Objective: Faster, More Accurate, and Even More Black Box](https://mlsanity.com/paper/2304.05527/) (2023)
- [Recursive Monte Carlo and Variational Inference with Auxiliary Variables](https://mlsanity.com/paper/2203.02836/) (2022)
- [Variational Marginal Particle Filters](https://mlsanity.com/paper/2109.15134/) (2021)
- [eVAE: Evolutionary Variational Autoencoder](https://mlsanity.com/paper/2301.00011/) (2023)
- [Sample Average Approximation for Black-Box VI](https://mlsanity.com/paper/2304.06803/) (2023)
- [Variational Inference with Locally Enhanced Bounds for Hierarchical Models](https://mlsanity.com/paper/2203.04432/) (2022)
- [Uphill Roads to Variational Tightness: Monotonicity and Monte Carlo Objectives](https://mlsanity.com/paper/2201.10989/) (2022)
- [U-Statistics for Importance-Weighted Variational Inference](https://mlsanity.com/paper/2302.13918/) (2023)
- [Variational Rejection Particle Filtering](https://mlsanity.com/paper/2103.15343/) (2021)
- [Towards Model-Agnostic Posterior Approximation for Fast and Accurate Variational Autoencoders](https://mlsanity.com/paper/2403.08941/) (2024)
- [Massively Parallel Expectation Maximization For Approximate Posteriors](https://mlsanity.com/paper/2503.08264/) (2025)
- [Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models](https://mlsanity.com/paper/2605.20345/) (2026)
- [Large-scale empirical tuning and comparison of default optimizers for variational inference](https://mlsanity.com/paper/2606.07841/) (2026)

## Abstract (arXiv)

Recent work in variational inference (VI) uses ideas from Monte Carlo estimation to tighten the lower bounds on the log-likelihood that are used as objectives. However, there is no systematic understanding of how optimizing different objectives relates to approximating the posterior distribution. Developing such a connection is important if the ideas are to be applied to inference-i.e., applications that require an approximate posterior and not just an approximation of the log-likelihood. Given a VI objective defined by a Monte Carlo estimator of the likelihood, we use a "divide and couple" procedure to identify augmented proposal and target distributions. The divergence between these is equal to the gap between the VI objective and the log-likelihood. Thus, after maximizing the VI objective, the augmented variational distribution may be used to approximate the posterior distribution.