###### Abstract
Since prognostics deals with the prediction of the future evolution of the system's health, there are several sources of potential uncertainty (e.g., inputs, states, parameters and mathematical models) that might affect these predictions. Therefore, it is essential to address these sources separately and to quantify how they influence the system in focus by capturing the overall uncertainty in the predictions. Using a probabilistic framework, the whole data-generating process is expressed in terms of probability distributions. The basic concept of probabilistic, or, synonymously, Bayesian inference is to update prior knowledge about unobserved quantities via the  
Figure 1: Structure of a typical LED system and its main thermal resistances determining the heat flow from junction to ambient. likelihood distribution into the posterior distribution. The posterior distribution then reflects the updated knowledge after having observed the data and provides a natural framework for UQ. This allows quantification of the reliability of model predictions by means of credible regions and posterior predictive distributions; cf. [19; 20; 21]. Bayesian hierarchical models (BMMs) provide a powerful and flexible probabilistic framework for quantification and analysis of multiple sources of uncertainty [22; 23; 24; 25]. The particular modelling of prior information in terms of several conditional prior distributions allows to account for different types of uncertainty in input parameters at different levels of the hierarchy and to differentiate between aleatory and epistemic uncertainty therein [21]. In this paper, a BHM framework is proposed to quantify the overall uncertainty in the initial LED temperature change caused by thermal stress. Its variability rests upon various sources, such as measurement and operating uncertainty and the LED's operational thermal resistance, which is subject to system and application variability. The BHM is built upon an inverse problem formulation of a physically derived model used to describe the initial temperature change. Posterior inference for this BHM is based on Markov chain Monte Carlo (MCMC) methods using a Metropolis-Hastings (MH) algorithm. Since temperature change, and particularly the initial temperature change, is a crucial factor for LED degradation due to thermal fatigue, its reliable quantification is decisive for the stability of lifetime predictions. Thus, accounting for different potential types and sources of uncertainty in the initial state is an important aspect in PC and the novelty of our approach. In an application scenario, we show how the estimated overall uncertainty in the starting temperature change can be used as valuable information in existing lifetime models to improve the reliability and accuracy of LED lifetime predictions. These models often assume fixed input parameters, and potential input uncertainty is not considered, leading to unstable predictions of the remaining long-term service live left. Beyond uncertainty quantification, one has to deal with the uncertainty in input conditions to reduce the uncertainty of the RUL. Thus, the need for proper uncertainty management arises to aid decision-making under uncertainty, which is one of the overall goals in PHM. The rest of the paper is organized as follows. Related work on probabilistic lifetime modelling of LED systems is discussed in Section 2, as well as on the Bayesian hierarchical modelling approach we will ultimately use. Section 3 introduces the BHM framework and its main model components, and outlines the MCMC sampling scheme for posterior inference. Different experimental scenarios for UQ in LED operation are considered in Section 4 based on simulated data. Results for these LED test data are presented in Section 5 and their potential effect on RUL prediction is discussed in Section 6 in an application scenario based on real LED measurement data. Finally, Section 7 gives some concluding remarks.