# Probabilistic Bayesian optimal experimental design using conditional normalizing flows

arXiv: https://arxiv.org/abs/2402.18337 · HTML: https://mlsanity.com/paper/2402.18337/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 20 references that are in the map.
- First posted on arXiv 28 Feb 2024.

## Builds on (most cited first, 16 of 16)

- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Variational Bayesian Optimal Experimental Design](https://mlsanity.com/paper/1903.05480/) (2019)
- [A Unified Stochastic Gradient Approach to Designing Bayesian-Optimal Experiments](https://mlsanity.com/paper/1911.00294/) (2019)
- [Learning-based Optimization of the Under-sampling Pattern in MRI](https://mlsanity.com/paper/1901.01960/) (2019)
- [Extending LOUPE for K-space Under-sampling Pattern Optimization in Multi-coil MRI](https://mlsanity.com/paper/2007.14450/) (2020)
- [On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows](https://mlsanity.com/paper/2402.06578/) (2024)
- [End-to-End Sequential Sampling and Reconstruction for MRI](https://mlsanity.com/paper/2105.06460/) (2021)
- [Edge-promoting adaptive Bayesian experimental design for X-ray imaging](https://mlsanity.com/paper/2104.00301/) (2021)
- [Design Amortization for Bayesian Optimal Experimental Design](https://mlsanity.com/paper/2210.03283/) (2022)
- [InvertibleNetworks.jl: A Julia package for scalable normalizing flows](https://mlsanity.com/paper/2312.13480/) (2023)
- [Sequential Experimental Design for X-Ray CT Using Deep Reinforcement Learning](https://mlsanity.com/paper/2307.06343/) (2023)

## Built on by (most cited first, 10 of 10)

- [Variational Bayesian Optimal Experimental Design with Normalizing Flows](https://mlsanity.com/paper/2404.13056/) (2024)
- [An uncertainty-aware Digital Shadow for underground multimodal CO2 storage monitoring](https://mlsanity.com/paper/2410.01218/) (2024)
- [JADAI: Jointly Amortizing Adaptive Design and Bayesian Inference](https://mlsanity.com/paper/2512.22999/) (2025)
- [Optimizing Likelihoods via Mutual Information: Bridging Simulation-Based Inference and Bayesian Optimal Experimental Design](https://mlsanity.com/paper/2502.08004/) (2025)
- [Sequential Bayesian Optimal Experimental Design in Infinite Dimensions via Policy Gradient Reinforcement Learning](https://mlsanity.com/paper/2601.05868/) (2026)
- [BEACON: Bayesian Experimental design Acceleration with Conditional Normalizing flows - a case study in optimal monitor well placement for CO2 sequestration](https://mlsanity.com/paper/2404.00075/) (2024)
- [Active MRI Acquisition with Diffusion Guided Bayesian Experimental Design](https://mlsanity.com/paper/2506.16237/) (2025)
- [Accelerated Bayesian Optimal Experimental Design via Conditional Density Estimation and Informative Data](https://mlsanity.com/paper/2507.15235/) (2025)
- [Approximation of differential entropy in Bayesian optimal experimental design](https://mlsanity.com/paper/2510.00734/) (2025)
- [Neural Optimal Design of Experiment for Inverse Problems](https://mlsanity.com/paper/2512.23763/) (2025)

## Abstract (arXiv)

Bayesian optimal experimental design (OED) seeks to conduct the most informative experiment under budget constraints to update the prior knowledge of a system to its posterior from the experimental data in a Bayesian framework. Such problems are computationally challenging because of (1) expensive and repeated evaluation of some optimality criterion that typically involves a double integration with respect to both the system parameters and the experimental data, (2) suffering from the curse-of-dimensionality when the system parameters and design variables are high-dimensional, (3) the optimization is combinatorial and highly non-convex if the design variables are binary, often leading to non-robust designs. To make the solution of the Bayesian OED problem efficient, scalable, and robust for practical applications, we propose a novel joint optimization approach. This approach performs simultaneous (1) training of a scalable conditional normalizing flow (CNF) to efficiently maximize the expected information gain (EIG) of a jointly learned experimental design (2) optimization of a probabilistic formulation of the binary experimental design with a Bernoulli distribution. We demonstrate the performance of our proposed method for a practical MRI data acquisition problem, one of the most challenging Bayesian OED problems that has high-dimensional (320 × 320) parameters at high image resolution, high-dimensional (640 × 386) observations, and binary mask designs to select the most informative observations.