Likelihood-Free Inference with Nuisance Parameters via Normalizing Flows (arXiv:2609.10534)

Forum topic · 小凯 · 2026-09-11

Summary

This arXiv paper by Phil Assheton proposes a simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic (or a close approximation) even in the presence of nuisance parameters, requiring only a sample generator from the distribution of interest. The statistic is near-pivotal in the sense of minimum average KL-divergence between its p-values and the uniform distribution, and is expected to have good statistical power when its dimension matches the parameter dimension. The method can incorporate prior knowledge of group invariances such as translation and scale. Empirically, it recovers the one-sample t-test almost exactly, outperforms the Welch test in worst-case size over a constrained variance-ratio range, achieves good calibration on partial biserial correlations, and offers higher power and much faster performance than profile likelihood-ratio techniques on small-to-moderate samples. This is a translation and summary of the forum post originally published on zhichai.net.

Paper Overview

Field: Machine Learning Author: Phil Assheton Published: 2026-09-09 arXiv: 2609.10534

English Translation of Abstract

We present a simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic (or something close) in the presence of nuisance parameters, based only on a sample generator from the distribution of interest. We show that the statistic is near-pivotal in the sense of minimum average KL-divergence of its

\(p\)

-values versus uniform and we argue that it can be expected to have good power when the dimension of the statistic equals the dimension of the parameter. It is able to incorporate prior knowledge about group invariances such as translation and scale. It can discover the one-sample

\(t\)

-test almost exactly, outperforms the Welch test in terms of worst-case size over a constrained variance-ratio range and achieves good calibration on partial biserial correlations, while showing higher power (and being much faster) on small-to-moderate samples than profile likelihood-ratio techniques.

Key Highlights

A normalizing flow decomposition that reveals an approximate pivotal statistic under nuisance parameters, requiring only a generator of samples from the target distribution.
Near-pivotalness quantified via minimum average KL-divergence of p-values from uniformity; good power expected when statistic dimension equals parameter dimension.
Supports encoding of group invariance priors (translation, scale).
Empirical results: near-exact recovery of the one-sample t-test; better worst-case size than the Welch test over a constrained variance-ratio range; good calibration on partial biserial correlations; higher power and much faster than profile likelihood-ratio methods on small-to-moderate samples.

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