# Recursive Monte Carlo and Variational Inference with Auxiliary Variables

arXiv: https://arxiv.org/abs/2203.02836 · HTML: https://mlsanity.com/paper/2203.02836/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 48 references that are in the map.
- First posted on arXiv 5 Mar 2022.

## Builds on (most cited first, 23 of 23)

- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)
- [Hierarchical Variational Models](https://mlsanity.com/paper/1511.02386/) (2015)
- [Variational inference for Monte Carlo objectives](https://mlsanity.com/paper/1602.06725/) (2016)
- [An Introduction to Probabilistic Programming](https://mlsanity.com/paper/1809.10756/) (2018)
- [Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives](https://mlsanity.com/paper/1810.04152/) (2018)
- [Reinterpreting Importance-Weighted Autoencoders](https://mlsanity.com/paper/1704.02916/) (2017)
- [Inference Compilation and Universal Probabilistic Programming](https://mlsanity.com/paper/1610.09900/) (2016)
- [Venture: a higher-order probabilistic programming platform with programmable inference](https://mlsanity.com/paper/1404.0099/) (2014)
- [Importance Weighted Hierarchical Variational Inference](https://mlsanity.com/paper/1905.03290/) (2019)
- [Nested Sequential Monte Carlo Methods](https://mlsanity.com/paper/1502.02536/) (2015)
- [Involutive MCMC: a Unifying Framework](https://mlsanity.com/paper/2006.16653/) (2020)
- [Divide and Couple: Using Monte Carlo Variational Objectives for Posterior Approximation](https://mlsanity.com/paper/1906.10115/) (2019)
- [Nested Variational Inference](https://mlsanity.com/paper/2106.11302/) (2021)
- [PClean: Bayesian Data Cleaning at Scale with Domain-Specific Probabilistic Programming](https://mlsanity.com/paper/2007.11838/) (2020)
- [Denotational validation of higher-order Bayesian inference](https://mlsanity.com/paper/1711.03219/) (2017)
- [A general perspective on the Metropolis-Hastings kernel](https://mlsanity.com/paper/2012.14881/) (2020)
- [Learning Proposals for Probabilistic Programs with Inference Combinators](https://mlsanity.com/paper/2103.00668/) (2021)
- [AIDE: An algorithm for measuring the accuracy of probabilistic inference algorithms](https://mlsanity.com/paper/1705.07224/) (2017)
- [Amortized Rejection Sampling in Universal Probabilistic Programming](https://mlsanity.com/paper/1910.09056/) (2019)
- [Composing inference algorithms as program transformations](https://mlsanity.com/paper/1603.01882/) (2016)
- [Estimators of Entropy and Information via Inference in Probabilistic Models](https://mlsanity.com/paper/2202.12363/) (2022)

## Built on by (most cited first, 8 of 8)

- [Sequential Monte Carlo Steering of Large Language Models using Probabilistic Programs](https://mlsanity.com/paper/2306.03081/) (2023)
- [Syntactic and Semantic Control of Large Language Models via Sequential Monte Carlo](https://mlsanity.com/paper/2504.13139/) (2025)
- [Fast Controlled Generation from Language Models with Adaptive Weighted Rejection Sampling](https://mlsanity.com/paper/2504.05410/) (2025)
- [Infinite Ends from Finite Samples: Open-Ended Goal Inference as Top-Down Bayesian Filtering of Bottom-Up Proposals](https://mlsanity.com/paper/2407.16770/) (2024)
- [Sequential Monte Carlo Learning for Time Series Structure Discovery](https://mlsanity.com/paper/2307.09607/) (2023)
- [Probabilistic Programming with Programmable Variational Inference](https://mlsanity.com/paper/2406.15742/) (2024)
- [Variational Nonlinear Kalman Filtering with Unknown Process Noise Covariance](https://mlsanity.com/paper/2305.03914/) (2023)
- [Particle GFlowNets: Rethinking Generative Marginalization Models](https://mlsanity.com/paper/2609.11538/) (2026)

## Abstract (arXiv)

A key design constraint when implementing Monte Carlo and variational inference algorithms is that it must be possible to cheaply and exactly evaluate the marginal densities of proposal distributions and variational families. This takes many interesting proposals off the table, such as those based on involved simulations or stochastic optimization. This paper broadens the design space, by presenting a framework for applying Monte Carlo and variational inference algorithms when proposal densities cannot be exactly evaluated. Our framework, recursive auxiliary-variable inference (RAVI), instead approximates the necessary densities using meta-inference: an additional layer of Monte Carlo or variational inference, that targets the proposal, rather than the model. RAVI generalizes and unifies several existing methods for inference with expressive approximating families, which we show correspond to specific choices of meta-inference algorithm, and provides new theory for analyzing their bias and variance. We illustrate RAVI's design framework and theorems by using them to analyze and improve upon Salimans et al.'s Markov Chain Variational Inference, and to design a novel sampler for Dirichlet process mixtures, achieving state-of-the-art results on a standard benchmark dataset from astronomy and on a challenging datacleaning task with Medicare hospital data.