Bayesian Calibrated Significance Levels Applied to the Spectral Tilt and Hemispherical Asymmetry

TL;DR

This paper introduces a Bayesian method to evaluate the maximum support for new parameters in models, applying it to cosmic microwave background data to assess spectral index and hemispherical asymmetry.

Contribution

It presents a computationally efficient approach to map p-values onto upper bounds of Bayes factors, aiding model comparison without strong prior assumptions.

Findings

Odds for spectral index deviation from one are 49:1.

Odds for hemispherical asymmetry are 9:1.

Method helps determine necessity of new parameters in models.

Abstract

Bayesian model selection provides a formal method of determining the level of support for new parameters in a model. However, if there is not a specific enough underlying physical motivation for the new parameters it can be hard to assign them meaningful priors, an essential ingredient of Bayesian model selection. Here we look at methods maximizing the prior so as to work out what is the maximum support the data could give for the new parameters. If the maximum support is not high enough then one can confidently conclude that the new parameters are unnecessary without needing to worry that some other prior may make them significant. We discuss a computationally efficient means of doing this which involves mapping p-values onto upper bounds of the Bayes factor (or odds) for the new parameters. A p-value of 0.05 ($1.96\sigma$) corresponds to odds less than or equal to 5:2 which is below the `weak' support at best threshold. A p-value of 0.0003 ($3.6\sigma$) corresponds to odds of less than or equal to 150:1 which is the `strong' support at best threshold. Applying this method we find that the odds on the scalar spectral index being different from one are 49:1 at best. We also find that the odds that there is primordial hemispherical asymmetry in the cosmic microwave background are 9:1 at best.