ENFR: Evidence Networks with Flow-Matched References for Amortized Simulation-Based Model Inference

Abstract

The evidence, 𝒵 = p(x | ℳ), of a data-generating model ℳ is a powerful tool in Bayesian frameworks: it enables model inference, model misspecification tests, and goodness-of-fit assessment as an alternative to standard discrepancy statistics, e.g. χ². While versatile, the evidence is notoriously difficult to compute, even when the likelihood is available. In both likelihood-based and simulation-based inference, estimating 𝒵 provides an immediate solution to high-dimensional goodness-of-fit or model misspecification tests: using 𝒵(x) as the test statistic induces a minimum-volume level-α test. Such tests are essential to assess model fidelity, which is necessary to trust simulation-based parameter inference. Existing learned approaches largely target Bayes factors, which only measure evidence ratios. Instead, we provide accurate, fast, and amortized estimates of the absolute evidence. We present ENFR: Evidence Networks with Flow-matched References. Building on the Evidence Network framework of Jeffrey & Wandelt (2024), the proposed method first learns Bayes factors with respect to a flow-matched reference, from which the absolute evidence is recovered. We show improvement in estimation of 𝒵 over the flow-based baseline in all applications considered and the best performance of all density estimation methods considered in our benchmarks. We furthermore introduce and provide the proof for an architecture that jointly learns the individual evidences of a set of models {ℳ_i}. We demonstrate the performance and scaling behavior of these approaches on data from a variety of scientific applications.

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Cite

BibTeX RIS

BibTeX

@misc{lehman2026enfr,
  title = {{ENFR: Evidence Networks with Flow-Matched References for Amortized Simulation-Based Model Inference}},
  author = {Kai Lehman and Jameson Dong and Carolina Cuesta-Lazaro and Niall Jeffrey and Benjamin D. Wandelt and Francisco Villaescusa-Navarro and Sven Krippendorf and Jochen Weller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23825},
  eprint = {2610.23825},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23825},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - ENFR: Evidence Networks with Flow-Matched References for Amortized Simulation-Based Model Inference
AU  - Kai Lehman
AU  - Jameson Dong
AU  - Carolina Cuesta-Lazaro
AU  - Niall Jeffrey
AU  - Benjamin D. Wandelt
AU  - Francisco Villaescusa-Navarro
AU  - Sven Krippendorf
AU  - Jochen Weller
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23825
UR  - https://arxiv.org/abs/2610.23825
AB  - The evidence, 𝒵 = p(x | ℳ), of a data-generating model ℳ is a powerful tool in Bayesian frameworks: it enables model inference, model misspecification tests, and goodness-of-fit assessment as an alternative to standard discrepancy statistics, e.g. χ^2. While versatile, the evidence is notoriously difficult to compute, even when the likelihood is available. In both likelihood-based and simulation-based inference, estimating 𝒵 provides an immediate solution to high-dimensional goodness-of-fit or model misspecification tests: using 𝒵(x) as the test statistic induces a minimum-volume level-α test. Such tests are essential to assess model fidelity, which is necessary to trust simulation-based parameter inference. Existing learned approaches largely target Bayes factors, which only measure evidence ratios. Instead, we provide accurate, fast, and amortized estimates of the absolute evidence. We present ENFR: Evidence Networks with Flow-matched References. Building on the Evidence Network framework of Jeffrey & Wandelt (2024), the proposed method first learns Bayes factors with respect to a flow-matched reference, from which the absolute evidence is recovered. We show improvement in estimation of 𝒵 over the flow-based baseline in all applications considered and the best performance of all density estimation methods considered in our benchmarks. We furthermore introduce and provide the proof for an architecture that jointly learns the individual evidences of a set of models {ℳ_i}. We demonstrate the performance and scaling behavior of these approaches on data from a variety of scientific applications.
ER  -

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