An Introduction to Bayesian and Frequentist Simulation-Based Inference with Machine Learning

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.

Neural simulation-based inference enables parameter estimation for complex models, but typically requires the user to specify a simulator encoding a fixed model structure. We present a framework for joint model selection and parameter estimation that combines large language models for program synthesis with neural simulation-based inference. Given a natural language description of the system and data under investigation, an LLM proposes candidate simulator programs which are iteratively refined via feedback-driven mutation and evaluated using neural density estimation. The approach enables simulation-based inference over a pool of models, not just parameters within a fixed model. On benchmarks spanning deterministic dynamics, stochastic epidemic models, and dark matter substructure inference from gravitational-lensing images, the method identifies plausible model families from open-ended prompts, with accuracy that reflects the information content of the data and identifiability of candidate models.

Simulation-based inference (SBI) of latent parameters is often hindered by simulator misspecification, the mismatch between simulated and real-world observations caused by inherent modeling simplifications. RoPE, the recent state-of-the-art for robust SBI, addresses this through optimal transport between learned representations of real and simulated observations, but requires ground-truth parameter calibration pairs that are typically unavailable in the very settings where SBI is needed. What practitioners do have is unstructured side-information such as regime labels, instruction text, and policy bulletins. We propose Misspecification-Aware Simulation-Based Inference (MA-SBI), a calibration-free framework that turns this side-channel into a posterior correction. A learned corrector maps side-channel text to an observation-space shift applied before any pre-trained amortized posterior, requiring no retraining and no parameter ground-truth. Our main theorem bounds achievable bias reduction by the mutual information between misspecification and side-channel, with a non-vacuous constant that extends to all sub-Gaussian noise via Donsker-Varadhan. On hide-the-calibration benchmarks, MA-SBI with text alone matches the oracle posterior across 10 seeds and two backbones (TOST equivalence), while RoPE given more data does not. The two approaches are complementary: where misspecification is structural and recoverable from parameter pairs, RoPE dominates, as the theory predicts. A stochastic variant improves posterior-predictive log-likelihood on real COVID and OxCGRT epidemiological data, and correctly leaves the posterior unchanged on a well-specified cognitive-science corpus.