# Partially Exchangeable Networks and Architectures for Learning Summary Statistics in Approximate Bayesian Computation

arXiv: https://arxiv.org/abs/1901.10230 · HTML: https://mlsanity.com/paper/1901.10230/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2019.

## Built on by (most cited first, 12 of 12)

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- [Learning Robust Statistics for Simulation-based Inference under Model Misspecification](https://mlsanity.com/paper/2305.15871/) (2023)
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- [Sequential Neural Posterior and Likelihood Approximation](https://mlsanity.com/paper/2102.06522/) (2021)
- [Sensitivity-Aware Amortized Bayesian Inference](https://mlsanity.com/paper/2310.11122/) (2023)
- [Black-box Bayesian inference for economic agent-based models](https://mlsanity.com/paper/2202.00625/) (2022)
- [Learning Summary Statistics for Bayesian Inference with Autoencoders](https://mlsanity.com/paper/2201.12059/) (2022)
- [Robust and integrative Bayesian neural networks for likelihood-free parameter inference](https://mlsanity.com/paper/2102.06521/) (2021)
- [Fuse It or Lose It: Deep Fusion for Multimodal Simulation-Based Inference](https://mlsanity.com/paper/2311.10671/) (2023)
- [Simulation-based inference for stochastic nonlinear mixed-effects models with applications in systems biology](https://mlsanity.com/paper/2504.11279/) (2025)
- [It Just Takes Two: Scaling Amortized Inference to Large Sets](https://mlsanity.com/paper/2605.07972/) (2026)
- [Amortized ratio-estimation importance sampling and localized simulation-based calibration for intractable likelihoods](https://mlsanity.com/paper/2609.39712/) (2026)

## Abstract (arXiv)

We present a novel family of deep neural architectures, named partially exchangeable networks (PENs) that leverage probabilistic symmetries. By design, PENs are invariant to block-switch transformations, which characterize the partial exchangeability properties of conditionally Markovian processes. Moreover, we show that any block-switch invariant function has a PEN-like representation. The DeepSets architecture is a special case of PEN and we can therefore also target fully exchangeable data. We employ PENs to learn summary statistics in approximate Bayesian computation (ABC). When comparing PENs to previous deep learning methods for learning summary statistics, our results are highly competitive, both considering time series and static models. Indeed, PENs provide more reliable posterior samples even when using less training data.