Splitting strategies for post-selection inference

Abstract

SUMMARY We consider the problem of providing valid inference for a selected parameter in a sparse regression setting. It is well known that classical regression tools can be unreliable in this context due to the bias generated in the selection step. Many approaches have been proposed in recent years to ensure inferential validity. Here, we consider a simple alternative to data splitting based on randomizing the response vector, which allows for higher selection and inferential power than the former and is applicable with an arbitrary selection rule. We provide a theoretical and empirical comparison of both methods and derive a central limit theorem for the randomization approach. Our investigations show that the gain in power can be substantial.

Generalized Data Thinning Using Sufficient Statistics

“…θ ? Some initial answers to these questions can be found in Neufeld et al (2023) and Rasines & Young (2022).…”

“…Then, U ∼ N n (θ, (1+γ)I n ) and V ∼ N n (θ, (1+γ −1 )I n ) are independent. Rasines & Young (2022) and Leiner et al (2022) applied this decomposition to address Scenario 1 in Section 1. Additionally, Rasines & Young (2022) showed that this leads to asymptotically valid inference under certain regularity conditions, even when X is not normally distributed.…”

“…Rasines & Young (2022) and Leiner et al (2022) applied this decomposition to address Scenario 1 in Section 1. Additionally, Rasines & Young (2022) showed that this leads to asymptotically valid inference under certain regularity conditions, even when X is not normally distributed. Tian (2020) and Oliveira et al (2021) applied this decomposition to address Scenario 2 in Section 1.…”

“…We are not the first to propose generalizations of sample splitting. Inspired by Tian & Taylor (2018)'s use of randomized responses, Rasines & Young (2022) define what they call the (U, V )-decomposition, which injects independent noise W to create two independent random variables U = u(X, W ) and V = v(X, W ) that together are jointly sufficient for the unknown parameters. However, they do not describe how to perform a (U, V )-decomposition other than in the special case of a Gaussian random vector with known covariance.…”

“…When these two random variables are independent (which they describe as the "P1" property), their proposal aligns with generalized thinning. However, like Rasines & Young (2022), they do not provide a general strategy for performing P1-fission, and the only two examples they provide are the Gaussian vector with known covariance and the Poisson.…”