Low-Budget Simulation-Based Inference with Bayesian Neural Networks

TL;DR

This work tackles calibration challenges in simulation-based inference (SBI) under data-poor regimes by introducing Bayesian neural networks with a principled functional prior. The authors design a Gaussian-process-based functional prior over posterior functions, map it to a weight prior, and show that a calibrated Bayesian model average emerges a priori, enabling reliable posterior estimation even with as few as $O(10)$ simulations. Empirical results on synthetic benchmarks and a cosmology N-body problem demonstrate that BNNs with the GP-informed prior produce well-calibrated, conservative posteriors and quantify epistemic uncertainty, outperforming standard SBI approaches at low budgets. The approach offers a practical path to reliable inference when simulators are expensive, with clear guidance on prior design, uncertainty decomposition, and applicability to high-cost scientific domains like cosmology.

Abstract

Simulation-based inference methods have been shown to be inaccurate in the data-poor regime, when training simulations are limited or expensive. Under these circumstances, the inference network is particularly prone to overfitting, and using it without accounting for the computational uncertainty arising from the lack of identifiability of the network weights can lead to unreliable results. To address this issue, we propose using Bayesian neural networks in low-budget simulation-based inference, thereby explicitly accounting for the computational uncertainty of the posterior approximation. We design a family of Bayesian neural network priors that are tailored for inference and show that they lead to well-calibrated posteriors on tested benchmarks, even when as few as $O(10)$ simulations are available. This opens up the possibility of performing reliable simulation-based inference using very expensive simulators, as we demonstrate on a problem from the field of cosmology where single simulations are computationally expensive. We show that Bayesian neural networks produce informative and well-calibrated posterior estimates with only a few hundred simulations.

Figures (13)

• Figure 1: Visualization of the prior tuned to match the GP prior on the SLCP benchmark. Left: examples of posterior functions sampled from the tuned prior over neural network's weights. Right: expected coverage of the prior Bayesian model average with the tuned prior and normal priors for varying standard deviations.
• Figure 2: Comparison of different simulation-based inference methods through the nominal log probability and coverage area under the curve. The higher the nominal log probability, the more performant the method is. A calibrated posterior approximation exhibits a coverage AUC of $0$. A positive coverage AUC indicates conservativeness, and a negative coverage AUC indicates overconfidence. 3 runs are performed, and the median is reported.
• Figure 3: Examples of $95 \%$ highest posterior density regions obtained with various algorithms and simulation budgets on the SLCP benchmark for a single observation. The black star represents the ground truth used to generate the observation.
• Figure 4: Comparison of posterior approximations obtained using a prior tuned to match the Gaussian process-based prior and using independent normal priors on weights with zero means and various standard deviations on the SLCP benchmark. 3 runs are performed, and the median is reported.
• Figure 5: Quantification of the different forms of uncertainties captured by the different NPE-based methods on the SLCP benchmark. 3 runs are performed, and the median is reported.

Theorems & Definitions (2)

• Proposition 1
• proof