# Adaptative significance levels in linear regression models with known variance

## TL;DR

This paper proposes a method to determine adaptive significance levels in Bayesian linear regression models with known variance, optimizing the evidence threshold in the FBST to balance error probabilities as sample size varies.

## Contribution

It introduces a novel approach to set adaptive cut-off values for the FBST in linear regression with known variance, accounting for sample size effects.

## Key findings

- Method minimizes combined type I and II errors for given sample sizes.
- Adaptive significance levels improve decision accuracy in Bayesian hypothesis testing.
- The approach provides a systematic way to choose evidence thresholds based on sample size.

## Abstract

The Full Bayesian Significance Test (FBST) for precise hypotheses was presented by Pereira and Stern [Entropy 1(4) (1999) 99-110] as a Bayesian alternative instead of the traditional significance test using p-value. The FBST is based on the evidence in favor of the null hypothesis (H). An important practical issue for the implementation of the FBST is the determination of how large the evidence must be in order to decide for its rejection. In the Classical significance tests, it is known that p-value decreases as sample size increases, so by setting a single significance level, it usually leads H rejection. In the FBST procedure, the evidence in favor of H exhibits the same behavior as the p-value when the sample size increases. This suggests that the cut-off point to define the rejection of H in the FBST should be a sample size function. In this work, the scenario of Linear Regression Models with known variance under the Bayesian approach is considered, and a method to find a cut-off value for the evidence in the FBST is presented by minimizing the linear combination of the averaged type I and type II error probabilities for a given sample size and also for a given dimension of the parametric space.