# Scalable Semi-Modular Inference with Variational Meta-Posteriors

arXiv: https://arxiv.org/abs/2204.00296 · HTML: https://mlsanity.com/paper/2204.00296/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 56 references that are in the map.
- First posted on arXiv 1 Apr 2022.

## Builds on (most cited first, 30 of 31)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC](https://mlsanity.com/paper/1507.04544/) (2015)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [Assigning a value to a power likelihood in a general Bayesian model](https://mlsanity.com/paper/1701.08515/) (2017)
- [Advances in Black-Box VI: Normalizing Flows, Importance Weighting, and Optimization](https://mlsanity.com/paper/2006.10343/) (2020)
- [A comparison of learning rate selection methods in generalized Bayesian inference](https://mlsanity.com/paper/2012.11349/) (2020)
- [Variational Bayes under Model Misspecification](https://mlsanity.com/paper/1905.10859/) (2019)
- [Robust, Accurate Stochastic Optimization for Variational Inference](https://mlsanity.com/paper/2009.00666/) (2020)
- [Better together? Statistical learning in models made of modules](https://mlsanity.com/paper/1708.08719/) (2017)
- [Semi-Modular Inference: enhanced learning in multi-modular models by tempering the influence of components](https://mlsanity.com/paper/2003.06804/) (2020)
- [Modularized Bayesian analyses and cutting feedback in likelihood-free inference](https://mlsanity.com/paper/2203.09782/) (2022)
- [Calibrating generalized predictive distributions](https://mlsanity.com/paper/2107.01688/) (2021)
- [Variational inference for cutting feedback in misspecified models](https://mlsanity.com/paper/2108.11066/) (2021)
- [Joining and splitting models with Markov melding](https://mlsanity.com/paper/1607.06779/) (2016)
- [Interoperability of statistical models in pandemic preparedness: principles and reality](https://mlsanity.com/paper/2109.13730/) (2021)
- [Using prior expansions for prior-data conflict checking](https://mlsanity.com/paper/1902.10393/) (2019)
- [Stochastic Approximation Cut Algorithm for Inference in Modularized Bayesian Models](https://mlsanity.com/paper/2006.01584/) (2020)
- [Asymptotics of cut distributions and robust modular inference using Posterior Bootstrap](https://mlsanity.com/paper/2110.11149/) (2021)
- [Valid belief updates for prequentially additive loss functions arising in Semi-Modular Inference](https://mlsanity.com/paper/2201.09706/) (2022)

## Built on by (most cited first, 2 of 2)

- [Amortized Simulation-Based Inference in Generalized Bayes via Neural Posterior Estimation](https://mlsanity.com/paper/2601.22367/) (2026)
- [Scalable and adaptive variational Bayes methods for Hawkes processes](https://mlsanity.com/paper/2212.00293/) (2022)

## Abstract (arXiv)

The Cut posterior and related Semi-Modular Inference are Generalised Bayes methods for Modular Bayesian evidence combination. Analysis is broken up over modular sub-models of the joint posterior distribution. Model-misspecification in multi-modular models can be hard to fix by model elaboration alone and the Cut posterior and SMI offer a way round this. Information entering the analysis from misspecified modules is controlled by an influence parameter η related to the learning rate. This paper contains two substantial new methods. First, we give variational methods for approximating the Cut and SMI posteriors which are adapted to the inferential goals of evidence combination. We parameterise a family of variational posteriors using a Normalising Flow for accurate approximation and end-to-end training. Secondly, we show that analysis of models with multiple cuts is feasible using a new Variational Meta-Posterior. This approximates a family of SMI posteriors indexed by η using a single set of variational parameters.