Use of Bayesian inference to diagnose issues in experimental measurements of mechanical disk resonators

Gravitational-wave detectors, such as LIGO, are predominantly limited by coating Brownian thermal noise (CTN), which arises from mechanical losses in the Bragg mirror coatings of test-mass optics. Accurately characterising and minimising these losses is crucial for enhancing detector sensitivity. This paper introduces a Bayesian framework for precisely analysing mechanical ring-down measurements of disk resonators, a standard method for quantifying mechanical loss in coating materials. Our approach incorporates an improved model that captures the non-linear behaviour of beam spot motion on split photodiode sensors, significantly improving upon traditional simplified exponential-decay methods, and models the non-stationary noise present in the data. We perform analyses using nested sampling and show, using Bayesian model comparison, that our improved model is strongly preferred over the existing model. We obtain more accurate estimates of the decay constants (τ1 and τ2), particularly for measurements with large oscillation amplitudes and for signals dominated by a single decay mode. This leads to improved accuracy in the measured losses: the uncertainty decreases by a factor of 2.9 for and 3.8 for , whilst the median change in the inferred values is less than 3% for both losses. Our method can also reliably analyze measurements that were previously discarded due to fitting errors, recovering 15% more usable measurements from our dataset, and can robustly characterise measurements where only a single decay is measured. This enhanced analytical capability yields more precise and reliable loss estimates from disk resonator data and provides a diagnostic tool for systematic errors, thereby supporting efforts to reduce coating thermal noise and improve gravitational-wave detector sensitivity.