Modeling Sampling Distributions of Test Statistics Using Autograd and Neural Networks

Core Concepts

Autograd can be used to take the derivative of neural network models of cumulative distribution functions (cdfs) to obtain approximations of the corresponding probability density functions (pdfs) of test statistics, which is useful for simulation-based frequentist inference.

Abstract

The authors explore the use of autograd to take the derivative of neural network models of cdfs in order to obtain approximations of the corresponding pdfs of test statistics. This is motivated by the fact that simulation-based frequentist inference methods require accurate modeling of either the p-value function or the cdf of the test statistic. The paper first considers the classic ON/OFF problem in astronomy and high-energy physics as a benchmark example. It is found that the ALFFI algorithm, which models the cdf as the mean of a certain discrete random variable, does not yield a sufficiently accurate smooth model of the cdf, and hence the derived pdf exhibits sharp fluctuations. The authors then directly model the empirical cdf as a function of the model parameters and the test statistic, which yields much better results. Conformal inference is used to quantify the uncertainty in the cdf and pdf models. The insights gained from the ON/OFF example are then applied to the SIR model in epidemiology, which exemplifies the utility of the methods for inference with intractable statistical models. Various techniques for uncertainty quantification, including Bayesian neural networks and bootstrap, are explored and compared. The paper concludes that directly modeling the empirical cdf is a viable approach, and that conformal inference provides a simple benchmark for calibrating other uncertainty quantification methods.

Stats

The authors generate synthetic data for the ON/OFF problem and the SIR model to train and evaluate their neural network models.

Quotes

"Autograd does not use finite difference approximations." "Strictly speaking, Eq. (1) applies only if x is from a continuous set. However, discrete distributions are frequently encountered in high-energy physics and other fields, and are often approximated by continuous distributions through suitable coarse-graining of x." "The key issue is whether a sufficiently accurate model of the cdf F(x | θ) can be constructed."

Key Insights Distilled From

Modelling Sampling Distributions of Test Statistics with Autograd

by Ali Al Kadhi... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02488.pdf

Deeper Inquiries

How can the conformal inference approach be extended to provide adaptive confidence intervals that are not only dependent on the model parameters θ, but also on the test statistic λ?

Conformal inference is a powerful method for constructing confidence intervals that guarantee marginal coverage over the population from which the dataset is sampled. To extend this approach to provide adaptive confidence intervals that depend not only on the model parameters θ but also on the test statistic λ, several modifications can be considered: Adaptive Calibration Sets: Instead of generating calibration sets only at specific parameter points, calibration sets can be created at various combinations of (θ, λ) values. This would allow for adaptive confidence intervals that adjust based on both the model parameters and the test statistic. Interpolation of Confidence Intervals: Develop a method to interpolate the width of the confidence intervals based on the values of both θ and λ. By creating a smooth interpolation function, confidence intervals can be adaptive to changes in both the model parameters and the test statistic. Conditional Coverage Guarantee: Aim to achieve conditional coverage guarantees, where the confidence intervals are not only adaptive to θ and λ but also provide coverage specific to the true parameter values. This would involve a more complex calibration process but would result in more accurate and tailored confidence intervals. Incorporating Prior Information: If prior information is available about the relationship between θ and λ, this information can be integrated into the conformal inference framework to create adaptive confidence intervals that reflect the joint distribution of θ and λ. By implementing these extensions, the conformal inference approach can be enhanced to provide adaptive confidence intervals that consider both the model parameters θ and the test statistic λ, offering more nuanced and accurate uncertainty quantification in statistical modeling.

How can the insights gained from this work be applied to other areas of computational science and engineering where accurate modeling of sampling distributions is important, such as in uncertainty quantification for complex simulations?

The insights gained from this work on modeling sampling distributions using neural networks and autograd can be applied to various areas of computational science and engineering where accurate uncertainty quantification is crucial. Some applications include: Complex Simulations: In fields like computational fluid dynamics, climate modeling, and structural analysis, accurate modeling of sampling distributions is essential for uncertainty quantification. By leveraging neural networks and autograd, researchers can develop robust methods for approximating sampling distributions and deriving confidence intervals for simulation outputs. Machine Learning: In machine learning applications, understanding the sampling distributions of model predictions is vital for assessing model performance and making reliable decisions. The techniques explored in this work can be adapted to improve uncertainty estimation in machine learning models, especially in high-dimensional settings. Risk Assessment: In risk assessment and financial modeling, accurate quantification of uncertainties is critical for decision-making. By applying the methods developed in this study, practitioners can enhance their ability to model and quantify uncertainties in risk analysis and financial forecasting. Optimization and Control: Uncertainty quantification plays a significant role in optimization problems and control systems. By incorporating advanced modeling techniques for sampling distributions, engineers can better assess the robustness and reliability of optimization algorithms and control strategies. Overall, the methodologies and approaches discussed in this work can be generalized and applied to a wide range of computational science and engineering domains where precise modeling of sampling distributions is fundamental for accurate decision-making and risk assessment.