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Engineering & Technology

Likelihood-Free Inference with Nuisance Parameters Using Normalizing Flow Decomposition

Research introduces a method for likelihood-free inference that addresses nuisance parameters through a decomposition of a neural-network-based normalizing flow

Overview

Research introduces a method for likelihood-free inference that addresses nuisance parameters through a decomposition of a neural-network-based normalizing flow. This decomposition aims to identify a statistic that is either pivotal or nearly pivotal, relying solely on a sample generator derived from the distribution under investigation. The approach offers a framework for inferring parameters when direct likelihood functions are intractable or unavailable.

Research Context

The core challenge addressed by this work is performing inference in models where the likelihood function is complex or cannot be computed analytically, especially when nuisance parameters are present. Nuisance parameters are those not directly of interest but whose values are necessary for evaluating the statistical model. Likelihood-free inference methods, such as Approximate Bayesian Computation (ABC), typically bypass explicit likelihood evaluations by simulating data from the model and comparing it to observed data. The presented method seeks to enhance this by leveraging normalizing flows to construct a statistic that simplifies inference by mitigating the influence of nuisance parameters.

Approach

The methodology involves a specific decomposition of a neural-network-based normalizing flow. This decomposition is designed to naturally uncover a statistic that exhibits properties of a pivotal statistic, or closely approximates one. A pivotal statistic is a function of observed data and unknown parameters whose distribution does not depend on the parameters. The construction of this statistic is based entirely on a sample generator, which produces data from the distribution of interest. This means the method does not require an explicit likelihood function.

The statistic is characterized as “near-pivotal” based on the minimum average Kullback-Leibler (KL) divergence of its $p$-values compared to a uniform distribution. This metric suggests that the distribution of $p$-values derived from this statistic closely resembles a uniform distribution under the null hypothesis, which is a desirable property for statistical tests.

The approach incorporates prior knowledge, specifically regarding group invariances such as translation and scale. This allows the method to integrate structural information about the data or parameters, potentially leading to more robust or efficient inference.

Findings

  • The neural-network-based normalizing flow decomposition naturally uncovers a statistic identified as pivotal or near-pivotal in the presence of nuisance parameters.
  • The statistic's near-pivotal nature is quantified by the minimum average KL-divergence of its $p$-values versus a uniform distribution.
  • The method is able to incorporate prior knowledge concerning group invariances, specifically translation and scale.
  • It successfully rediscovered the one-sample $t$-test with high accuracy.
  • The method demonstrated superior performance compared to the Welch test in terms of worst-case size, specifically within a constrained variance-ratio range.
  • It achieved good calibration when applied to partial biserial correlations.
  • For small-to-moderate sample sizes, the method exhibited higher power and significantly faster operation compared to profile likelihood-ratio techniques. This suggests practical advantages in scenarios with limited data or computational time constraints.
  • The effectiveness of the statistic is expected to be high when the dimension of the statistic aligns with the dimension of the parameter.

Why This Matters

This research offers advancements in likelihood-free inference, providing a method that can operate effectively even when traditional likelihood functions are inaccessible or computationally prohibitive. The ability to automatically discover near-pivotal statistics and integrate group invariances represents a step forward in robust statistical modeling. The observed improvements in power and speed over existing methods, particularly for smaller sample sizes, indicate potential for broader applicability in scientific and engineering domains where data generation is complex or limited.

Key Findings

  • A simple decomposition of a neural-network-based normalizing flow uncovers a near-pivotal statistic for nuisance parameters.
  • The statistic's near-pivotal property is evidenced by minimum average KL-divergence of its p-values versus uniform.
  • The method can integrate prior knowledge about group invariances like translation and scale.
  • It almost exactly rediscovers the one-sample $t$-test.
  • It outperforms the Welch test in worst-case size over a constrained variance-ratio range.
  • Achieves good calibration on partial biserial correlations and higher power/speed than profile likelihood-ratio techniques for small-to-moderate samples.