On the relationship between a Gamma distributed precision parameter and the associated standard deviation in the context of Bayesian parameter inference

TL;DR

This paper presents a numerical optimization method to transform between a Gamma distributed precision parameter and the distribution of the associated standard deviation, facilitating Bayesian inference with prior information on measurement uncertainty.

Contribution

It introduces a novel method for converting between Gamma distributed precision and standard deviation distributions in Bayesian inference.

Findings

The method performs well across various scenarios.

It enables practical incorporation of prior information on measurement uncertainty.

The approach is based on numerical optimization techniques.

Abstract

In Bayesian inference, an unknown measurement uncertainty is often quantified in terms of a Gamma distributed precision parameter, which is impractical when prior information on the standard deviation of the measurement uncertainty shall be utilised during inference. This paper thus introduces a method for transforming between a gamma distributed precision parameter and the distribution of the associated standard deviation. The proposed method is based on numerical optimisation and shows adequate results for a wide range of scenarios.