# Mixed neural posterior estimation for simulators with discrete and continuous parameters

arXiv: https://arxiv.org/abs/2605.13551 · HTML: https://mlsanity.com/paper/2605.13551/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 64 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 30 of 34)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Measuring Calibration in Deep Learning](https://mlsanity.com/paper/1904.01685/) (2019)
- [TabDDPM: Modelling Tabular Data with Diffusion Models](https://mlsanity.com/paper/2209.15421/) (2022)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Evaluating model calibration in classification](https://mlsanity.com/paper/1902.06977/) (2019)
- [Benchmarking Simulation-Based Inference](https://mlsanity.com/paper/2101.04653/) (2021)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Validating Bayesian Inference Algorithms with Simulation-Based Calibration](https://mlsanity.com/paper/1804.06788/) (2018)
- [Mitigating Bias in Calibration Error Estimation](https://mlsanity.com/paper/2012.08668/) (2020)
- [Calibration tests in multi-class classification: A unifying framework](https://mlsanity.com/paper/1910.11385/) (2019)
- [Real-time gravitational-wave science with neural posterior estimation](https://mlsanity.com/paper/2106.12594/) (2021)
- [CoDi: Co-evolving Contrastive Diffusion Models for Mixed-type Tabular Synthesis](https://mlsanity.com/paper/2304.12654/) (2023)
- [TabDiff: a Mixed-type Diffusion Model for Tabular Data Generation](https://mlsanity.com/paper/2410.20626/) (2024)
- [Investigating the Impact of Model Misspecification in Neural Simulation-based Inference](https://mlsanity.com/paper/2209.01845/) (2022)
- [Truncated proposals for scalable and hassle-free simulation-based inference](https://mlsanity.com/paper/2210.04815/) (2022)
- [Simulation-Based Inference: A Practical Guide](https://mlsanity.com/paper/2508.12939/) (2025)
- [sbi reloaded: a toolkit for simulation-based inference workflows](https://mlsanity.com/paper/2411.17337/) (2024)
- [Amortized Bayesian model comparison with evidential deep learning](https://mlsanity.com/paper/2004.10629/) (2020)
- [HNPE: Leveraging Global Parameters for Neural Posterior Estimation](https://mlsanity.com/paper/2102.06477/) (2021)
- [DLT: Conditioned layout generation with Joint Discrete-Continuous Diffusion Layout Transformer](https://mlsanity.com/paper/2303.03755/) (2023)
- [Robust and Scalable Bayesian Online Changepoint Detection](https://mlsanity.com/paper/2302.04759/) (2023)
- [Amortized Bayesian Multilevel Models](https://mlsanity.com/paper/2408.13230/) (2024)
- [Effortless, Simulation-Efficient Bayesian Inference using Tabular Foundation Models](https://mlsanity.com/paper/2504.17660/) (2025)
- [Simulation-based Bayesian inference under model misspecification](https://mlsanity.com/paper/2503.12315/) (2025)

## Abstract (arXiv)

Neural Posterior Estimation (NPE) enables rapid parameter inference for complex simulators with intractable likelihoods. NPE trains an inference network to estimate a probability density over parameters given data, typically assumed to be continuous. However, many scientific models involve parameter spaces that are mixed, that is, they contain both discrete and continuous dimensions. We address this limitation by extending NPE to mixed parameter spaces through an inference network that jointly handles discrete and continuous parameters. The inference network factorizes the joint posterior into discrete and continuous components, combining an autoregressive classifier for the discrete parameters with a generative model for the continuous parameters, trained jointly under a single simulation-based objective. In addition, we propose a diagnostic tool to assess the calibration of the mixed posterior approximation. Across tractable toy examples and real-world scientific simulators, our joint inference approach yields accurate and calibrated posteriors. The inference framework is available in the textttsbi Python package.