# Compositional Score Modeling for Simulation-based Inference

arXiv: https://arxiv.org/abs/2209.14249 · HTML: https://mlsanity.com/paper/2209.14249/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 70 references that are in the map.
- First posted on arXiv 28 Sep 2022.

## Builds on (most cited first, 30 of 41)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Compositional Visual Generation with Composable Diffusion Models](https://mlsanity.com/paper/2206.01714/) (2022)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models](https://mlsanity.com/paper/2104.14951/) (2021)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [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)
- [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (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)

## Built on by (most cited first, 30 of 48)

- [All-in-one simulation-based inference](https://mlsanity.com/paper/2404.09636/) (2024)
- [Amortizing intractable inference in diffusion models for vision, language, and control](https://mlsanity.com/paper/2405.20971/) (2024)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Steering Masked Discrete Diffusion Models via Discrete Denoising Posterior Prediction](https://mlsanity.com/paper/2410.08134/) (2024)
- [Simulation-Based Inference: A Practical Guide](https://mlsanity.com/paper/2508.12939/) (2025)
- [Diffusion posterior sampling for simulation-based inference in tall data settings](https://mlsanity.com/paper/2404.07593/) (2024)
- [Amortized In-Context Bayesian Posterior Estimation](https://mlsanity.com/paper/2502.06601/) (2025)
- [Leveraging Self-Consistency for Data-Efficient Amortized Bayesian Inference](https://mlsanity.com/paper/2310.04395/) (2023)
- [Amortized Bayesian Multilevel Models](https://mlsanity.com/paper/2408.13230/) (2024)
- [Does learning the right latent variables necessarily improve in-context learning?](https://mlsanity.com/paper/2405.19162/) (2024)
- [Effortless, Simulation-Efficient Bayesian Inference using Tabular Foundation Models](https://mlsanity.com/paper/2504.17660/) (2025)
- [Compositional simulation-based inference for time series](https://mlsanity.com/paper/2411.02728/) (2024)
- [Simulation-based Inference for High-dimensional Data using Surjective Sequential Neural Likelihood Estimation](https://mlsanity.com/paper/2308.01054/) (2023)
- [Conditional diffusions for amortized neural posterior estimation](https://mlsanity.com/paper/2410.19105/) (2024)
- [Product of Experts for Visual Generation](https://mlsanity.com/paper/2506.08894/) (2025)
- [CP4SBI: Local Conformal Calibration of Credible Sets in Simulation-Based Inference](https://mlsanity.com/paper/2508.17077/) (2025)
- [Robust Simulation-Based Inference under Missing Data via Neural Processes](https://mlsanity.com/paper/2503.01287/) (2025)
- [Multilevel neural simulation-based inference](https://mlsanity.com/paper/2506.06087/) (2025)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (2025)
- [Zero-Flow Encoders](https://mlsanity.com/paper/2602.00797/) (2026)
- [Fuse It or Lose It: Deep Fusion for Multimodal Simulation-Based Inference](https://mlsanity.com/paper/2311.10671/) (2023)
- [Amortized Bayesian Workflow](https://mlsanity.com/paper/2409.04332/) (2024)
- [Posterior SBC: Simulation-Based Calibration Checking Conditional on Data](https://mlsanity.com/paper/2502.03279/) (2025)
- [PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inference](https://mlsanity.com/paper/2510.13763/) (2025)
- [Simulation-Based Inference of Surface Accumulation and Basal Melt Rates of an Antarctic Ice Shelf from Isochronal Layers](https://mlsanity.com/paper/2312.02997/) (2023)
- [Amortized Simulation-Based Inference in Generalized Bayes via Neural Posterior Estimation](https://mlsanity.com/paper/2601.22367/) (2026)
- [Synthesizing 3D Abstractions by Inverting Procedural Buildings with Transformers](https://mlsanity.com/paper/2501.17044/) (2025)
- [Diffusion Models for Inverse Problems in the Exponential Family](https://mlsanity.com/paper/2502.05994/) (2025)
- [Causal Posterior Estimation](https://mlsanity.com/paper/2505.21468/) (2025)
- [Conformal C2ST: Turning weak classifiers into strong two-sample tests](https://mlsanity.com/paper/2507.17026/) (2025)

## Abstract (arXiv)

Neural Posterior Estimation methods for simulation-based inference can be ill-suited for dealing with posterior distributions obtained by conditioning on multiple observations, as they tend to require a large number of simulator calls to learn accurate approximations. In contrast, Neural Likelihood Estimation methods can handle multiple observations at inference time after learning from individual observations, but they rely on standard inference methods, such as MCMC or variational inference, which come with certain performance drawbacks. We introduce a new method based on conditional score modeling that enjoys the benefits of both approaches. We model the scores of the (diffused) posterior distributions induced by individual observations, and introduce a way of combining the learned scores to approximately sample from the target posterior distribution. Our approach is sample-efficient, can naturally aggregate multiple observations at inference time, and avoids the drawbacks of standard inference methods.