## 3 Model and Methods
### Bayesian Hierarchical Modelling Framework
#### 3.2.2 Prior Distributions  
Bayesian model specification ends up with an entire probabilistic model in terms of a joint probability distribution for all observable and unobservable quantities in a problem. The uncertainty of the unobserved parameters \(\mathbf{\Theta}\) is modelled by the prior probability distribution \(p(\mathbf{\Theta})\), which allows incorporation of available prior information from various sources. We take a hierarchical modelling approach that allows to consider different types of physical input parameters in (4) which differ in their level and nature of uncertainty:  
1. Deterministic inputs: The parameters \(\eta_{h}\) and \(V_{f}\) in (4) are assumed to be perfectly known and are thus considered as deterministic input parameters, i.e., \(\mathbf{z}_{i}=(\eta_{h,i}=\eta_{h},V_{f,i}=V_{f})\). 2. Aleatory variability: In general, hierarchically specified prior distributions may be assigned to input parameters that are subject to aleatory variability. We classify these types of parameters into parameters for which (a) randomness is induced by known variability and (b) randomness is induced by variability that is itself unknown. Similar prior considerations are used and discussed in [PERSON] and Sudret [24] for simply supported beam deflections. 1. Known variability: We assume that the physically realized currents \(I_{f,i}\) in (4) (may) vary randomly between different systems due to some known degree of uncertainty that is inherent to the equipment used. Thus, the hyper-parameters \(\mathbf{\theta}_{I_{f,i}}\) of the prior \[I_{f,i}\sim p(I_{f,i}|\mathbf{\theta}_{I_{f,i}})\] (7) represent available information about the accuracy of the power unit adjusting \(I_{f,i}\). The uncertainty in the realized values \(I_{f,i}\) is then \"prescribed\" by the hyper-parameters \(\mathbf{\theta}_{I_{f,i}}\); cf. [24]. 2. Unknown variability: Variabilities in the thermal resistances \(R_{\text{th}_{i}}\) across identically manufactured LED systems may result either from production (e.g., due to irregularities in the manufacturing process of LED packages) or from the setup or application (e.g., due to the TIM). For both sources, the magnitude of variability is assumed to be unknown. For input parameters with unknown aleatory, the structure of the prior in (7) is extended by another level of hierarchy. We assume that the thermal resistances are subject to this type of \"unknown\" aleatory variability, and we specify the prior as \[R_{\text{th}_{i}} \sim p(R_{\text{th}_{i}}|\mathbf{\theta}_{R_{\text{th}}}),\] (8) \[\mathbf{\theta}_{R_{\text{th}}} \sim p(\mathbf{\theta}_{R_{\text{th}}}|\mathbf{\gamma}_{R_{\text{th}}}),\] (9) where the hyper-parameters \(\mathbf{\vartheta}_{R_{\text{th}}}\) are assigned a hyper-prior given in (9) rather than being fixed with additional hyper-parameters \(\mathbf{\gamma}_{R_{\text{th}}}\). 3. Epistemic uncertainty: The hyper-parameters \(\mathbf{\vartheta}_{R_{\text{th}}}\) in (9) refer to unknown parameters that are subject to epistemic uncertainty, which is modelled as \(\mathbf{\vartheta}_{R_{\text{th}}}\sim p(\mathbf{\vartheta}_{R_{\text{th}}}|\mathbf{ \gamma}_{R_{\text{th}}})\). In general, epistemic uncertainty is reducible in practice with increased knowledge, whereas the aleatory variability in (2a) and (2b) is not. Note also that \(\eta_{h}\) and \(V_{f}\) may be subject to variabilities in practice. Then, prior distributions as described in (2a) have to be specified. We denote by \(\mathbf{\Theta}=(\mathbf{R}_{\text{th}},\mathbf{\vartheta}_{R_{\text{th}}},\mathbf{I}_{f})\) the collection of all parameters and hyper-parameters for which a prior distribution has to be specified.