# Solving high-dimensional parameter inference: marginal posterior densities & Moment Networks

arXiv: https://arxiv.org/abs/2011.05991 · HTML: https://mlsanity.com/paper/2011.05991/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Nov 2020.

## Built on by (most cited first, 22 of 22)

- [Truncated Marginal Neural Ratio Estimation](https://mlsanity.com/paper/2107.01214/) (2021)
- [The Cosmic Graph: Optimal Information Extraction from Large-Scale Structure using Catalogues](https://mlsanity.com/paper/2207.05202/) (2022)
- [Fast and Credible Likelihood-Free Cosmology with Truncated Marginal Neural Ratio Estimation](https://mlsanity.com/paper/2111.08030/) (2021)
- [Multifield Cosmology with Artificial Intelligence](https://mlsanity.com/paper/2109.09747/) (2021)
- [Evidence Networks: simple losses for fast, amortized, neural Bayesian model comparison](https://mlsanity.com/paper/2305.11241/) (2023)
- [Contrastive Neural Ratio Estimation for Simulation-based Inference](https://mlsanity.com/paper/2210.06170/) (2022)
- [Robust marginalization of baryonic effects for cosmological inference at the field level](https://mlsanity.com/paper/2109.10360/) (2021)
- [Machine Learning and Cosmology](https://mlsanity.com/paper/2203.08056/) (2022)
- [Domain Adaptive Graph Neural Networks for Constraining Cosmological Parameters Across Multiple Data Sets](https://mlsanity.com/paper/2311.01588/) (2023)
- [Listening to the Noise: Blind Denoising with Gibbs Diffusion](https://mlsanity.com/paper/2402.19455/) (2024)
- [Multilevel neural simulation-based inference](https://mlsanity.com/paper/2506.06087/) (2025)
- [Field-level simulation-based inference with galaxy catalogs: the impact of systematic effects](https://mlsanity.com/paper/2310.15234/) (2023)
- [Simulation-Based Inference with Quantile Regression](https://mlsanity.com/paper/2401.02413/) (2024)
- [Simulation-based stacking](https://mlsanity.com/paper/2310.17009/) (2023)
- [Improving the Accuracy of Marginal Approximations in Likelihood-Free Inference via Localisation](https://mlsanity.com/paper/2207.06655/) (2022)
- [Optimal simulation-based Bayesian decisions](https://mlsanity.com/paper/2311.05742/) (2023)
- [How many simulations do we need for simulation-based inference in cosmology?](https://mlsanity.com/paper/2503.13755/) (2025)
- [Fishnets: Information-Optimal, Scalable Aggregation for Sets and Graphs](https://mlsanity.com/paper/2310.03812/) (2023)
- [Learnable wavelet neural networks for cosmological inference](https://mlsanity.com/paper/2307.14362/) (2023)
- [Cosmology with Persistent Homology: Parameter Inference via Machine Learning](https://mlsanity.com/paper/2412.15405/) (2024)
- [Reconstructing the local density field with combined convolutional and point cloud architecture](https://mlsanity.com/paper/2510.08573/) (2025)
- [Dark Energy Survey Year 3 results: Simulation-based wCDM inference from weak lensing and galaxy clustering maps with deep learning: Analysis design](https://mlsanity.com/paper/2511.04681/) (2025)

## Abstract (arXiv)

High-dimensional probability density estimation for inference suffers from the "curse of dimensionality". For many physical inference problems, the full posterior distribution is unwieldy and seldom used in practice. Instead, we propose direct estimation of lower-dimensional marginal distributions, bypassing high-dimensional density estimation or high-dimensional Markov chain Monte Carlo (MCMC) sampling. By evaluating the two-dimensional marginal posteriors we can unveil the full-dimensional parameter covariance structure. We additionally propose constructing a simple hierarchy of fast neural regression models, called Moment Networks, that compute increasing moments of any desired lower-dimensional marginal posterior density; these reproduce exact results from analytic posteriors and those obtained from Masked Autoregressive Flows. We demonstrate marginal posterior density estimation using high-dimensional LIGO-like gravitational wave time series and describe applications for problems of fundamental cosmology.