# Sequential Neural Posterior and Likelihood Approximation

arXiv: https://arxiv.org/abs/2102.06522 · HTML: https://mlsanity.com/paper/2102.06522/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 51 references that are in the map.
- First posted on arXiv 12 Feb 2021.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [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)
- [On Contrastive Learning for Likelihood-free Inference](https://mlsanity.com/paper/2002.03712/) (2020)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models](https://mlsanity.com/paper/1501.03291/) (2015)
- [Fast ε-free Inference of Simulation Models with Bayesian Conditional Density Estimation](https://mlsanity.com/paper/1605.06376/) (2016)
- [Partially Exchangeable Networks and Architectures for Learning Summary Statistics in Approximate Bayesian Computation](https://mlsanity.com/paper/1901.10230/) (2019)
- [Sequential Neural Methods for Likelihood-free Inference](https://mlsanity.com/paper/1811.08723/) (2018)

## Built on by (most cited first, 30 of 30)

- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks](https://mlsanity.com/paper/2112.08866/) (2021)
- [Compositional Score Modeling for Simulation-based Inference](https://mlsanity.com/paper/2209.14249/) (2022)
- [Learning Robust Statistics for Simulation-based Inference under Model Misspecification](https://mlsanity.com/paper/2305.15871/) (2023)
- [Variational methods for simulation-based inference](https://mlsanity.com/paper/2203.04176/) (2022)
- [JANA: Jointly Amortized Neural Approximation of Complex Bayesian Models](https://mlsanity.com/paper/2302.09125/) (2023)
- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [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)
- [Consistency Models for Scalable and Fast Simulation-Based Inference](https://mlsanity.com/paper/2312.05440/) (2023)
- [Sensitivity-Aware Amortized Bayesian Inference](https://mlsanity.com/paper/2310.11122/) (2023)
- [Generalized Bayesian Inference for Scientific Simulators via Amortized Cost Estimation](https://mlsanity.com/paper/2305.15208/) (2023)
- [Leveraging Self-Consistency for Data-Efficient Amortized Bayesian Inference](https://mlsanity.com/paper/2310.04395/) (2023)
- [Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference](https://mlsanity.com/paper/2301.11674/) (2023)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Multifidelity Simulation-based Inference for Computationally Expensive Simulators](https://mlsanity.com/paper/2502.08416/) (2025)
- [Misspecification-robust Sequential Neural Likelihood for Simulation-based Inference](https://mlsanity.com/paper/2301.13368/) (2023)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks: An Extended Investigation](https://mlsanity.com/paper/2406.03154/) (2024)
- [A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)](https://mlsanity.com/paper/2407.16550/) (2024)
- [Demonstrating the power and flexibility of variational assumptions for amortized neural posterior estimation in environmental applications](https://mlsanity.com/paper/2404.10899/) (2024)
- [Fuse It or Lose It: Deep Fusion for Multimodal Simulation-Based Inference](https://mlsanity.com/paper/2311.10671/) (2023)
- [Fast, accurate and lightweight sequential simulation-based inference using Gaussian locally linear mappings](https://mlsanity.com/paper/2403.07454/) (2024)
- [Improving the Accuracy of Marginal Approximations in Likelihood-Free Inference via Localisation](https://mlsanity.com/paper/2207.06655/) (2022)
- [Variational Autoencoders for Efficient Simulation-Based Inference](https://mlsanity.com/paper/2411.14511/) (2024)
- [Simulation-based inference for stochastic nonlinear mixed-effects models with applications in systems biology](https://mlsanity.com/paper/2504.11279/) (2025)
- [OneFlowSBI: One Model, Many Queries for Simulation-Based Inference](https://mlsanity.com/paper/2601.22951/) (2026)
- [Supercharging Simulation-Based Inference for Bayesian Optimal Experimental Design](https://mlsanity.com/paper/2602.06900/) (2026)
- [An Optimal Likelihood Free Method for Biological Model Selection](https://mlsanity.com/paper/2208.02344/) (2022)
- [Misspecification-robust amortised simulation-based inference using variational methods](https://mlsanity.com/paper/2509.05724/) (2025)
- [Amortized ratio-estimation importance sampling and localized simulation-based calibration for intractable likelihoods](https://mlsanity.com/paper/2609.39712/) (2026)

## Abstract (arXiv)

We introduce the sequential neural posterior and likelihood approximation (SNPLA) algorithm. SNPLA is a normalizing flows-based algorithm for inference in implicit models, and therefore is a simulation-based inference method that only requires simulations from a generative model. SNPLA avoids Markov chain Monte Carlo sampling and correction-steps of the parameter proposal function that are introduced in similar methods, but that can be numerically unstable or restrictive. By utilizing the reverse KL divergence, SNPLA manages to learn both the likelihood and the posterior in a sequential manner. Over four experiments, we show that SNPLA performs competitively when utilizing the same number of model simulations as used in other methods, even though the inference problem for SNPLA is more complex due to the joint learning of posterior and likelihood function. Due to utilizing normalizing flows SNPLA generates posterior draws much faster (4 orders of magnitude) than MCMC-based methods.