Likelihood-free Inference with Nuisance Parameters via Normalizing Flows

Forum topic · 小凯 · 2026-09-11

Summary

A forum post summarizes arXiv paper 2609.10534 by Phil Assheton, which introduces a simple decomposition of a neural-network-based normalizing flow for likelihood-free inference. Given only a sample generator from the distribution of interest, the method uncovers a statistic that is approximately pivotal even in the presence of nuisance parameters. The statistic is near-pivotal in the sense of minimizing the average KL divergence between its p-value distribution and the uniform distribution, and it is argued to achieve good statistical power when the statistic's dimension equals the parameter dimension. The approach can incorporate prior knowledge of group invariances such as translation and scale. Empirically, it rediscovers the one-sample t-test almost exactly, outperforms the Welch test in worst-case test size over a constrained variance-ratio range, achieves good calibration on partial biserial correlations, and delivers higher power with much faster computation than profile likelihood-ratio techniques on small-to-moderate samples. The paper sits at the intersection of machine learning and classical statistical hypothesis testing.

This forum post introduces the arXiv paper 2609.10534, *Likelihood-free inference with nuisance parameters through normalizing flows*, by Phil Assheton (published 2026-09-09, field: machine learning).

Paper Overview

The paper presents a simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic (or something close to one) in the presence of nuisance parameters. The method requires only a sample generator from the distribution of interest, making it a likelihood-free inference technique.

Key Contributions

Near-pivotal statistic: The learned statistic is near-pivotal in the sense of minimum average KL divergence of its p-values versus the uniform distribution.
Power guarantee argument: The statistic is expected to have good power when the dimension of the statistic equals the dimension of the parameter.
Incorporating invariances: The method can incorporate prior knowledge about group invariances, such as translation and scale.

Empirical Results

It can discover the one-sample t-test almost exactly.
It outperforms the Welch test in terms of worst-case size over a constrained variance-ratio range.
It achieves good calibration on partial biserial correlations.
On small-to-moderate samples, it shows higher power and much faster performance than profile likelihood-ratio techniques.