Simulation-Based Empirical Bayes

2026-07-23

AI Summary

This work addresses the challenge that in many scientific applications, likelihood functions are only accessible implicitly through simulators, rendering traditional empirical Bayes methods—which rely on explicit likelihoods—ineffective. To overcome this limitation, the authors propose Simulation-Based Empirical Bayes (SBEB), the first approach to extend empirical Bayes to implicit-likelihood settings. SBEB integrates nonparametric empirical Bayes, simulation-based inference (SBI), and amortized inference networks, iteratively optimizing to learn a prior from observed data and simulator-generated samples that approximates the true population prior. Experimental results demonstrate that SBEB significantly outperforms conventional SBI methods employing fixed priors across multiple scientific simulators and real-world datasets, yielding substantially improved parameter estimation accuracy.

Abstract

Empirical Bayes (EB) performs simultaneous inference across many related latent variables. Classical EB assumes that the likelihood p(x | z) is tractable. In many scientific applications, however, the likelihood is available only through a simulator. This paper develops EB for such implicit likelihoods. We introduce simulation-based empirical Bayes (SBEB), which connects nonparametric EB to simulation-based inference (SBI). SBEB computes EB estimates without an explicit density by using the observed data, simulator samples, and an amortized inference network. SBEB iteratively refines the fitted EB prior toward the population prior. With several scientific simulators and real-world data, we demonstrate that SBEB improves accuracy over SBI with a fixed prior.

Problem

• Empirical Bayes
• implicit likelihood
• simulation-based inference
• simulator
• nonparametric EB

Innovation

• Simulation-Based Inference
• Empirical Bayes
• Implicit Likelihood
• Amortized Inference
• Nonparametric Bayes