## 4 Simulation Experiments
### Test Datasets  
Different experiments are assigned to the problem setups above to show how these issues can be addressed in applications.  
**Experiment 1 (E1)**: In the default setting, the proportion of input power which remains as heat, i.e., \(I_{\text{heat}}=I_{f}\cdot V_{f}\cdot\eta_{h}\), is well-known. The LED systems are exposed to electric input power with forward current \(I_{f}=0.68\) A and forward voltage \(V_{f}=11.909\) V. The amountof input power that is converted to heat is assumed to be fixed at \(\eta_{h}=0.35\). The simulated LED systems are similar insofar that we assume variability in \(R_{\text{th}_{i}}\). Thus, for \(i=1,\ldots,n\), the total thermal resistances \(R_{\text{th}_{i}}\) are sampled from a lognormal distribution with parameters \(\mu_{R_{\text{th}_{i}}}=39.13\text{ K}/\text{W}\) and \(\sigma_{R_{\text{th}_{i}}}=1\text{ K}/\text{W}\), which corresponds to measurements of a (specimen) LED system in the experimental setup used in [PERSON] et al. (2015).  
For \(i=1,\ldots,n\), the temperature swing at starting, \(\tilde{y}_{i}=\Delta\tilde{T}_{0}\), is computed according to (2) and perturbed by a noise term \(\varepsilon_{i}\sim\mathcal{N}\big{(}0,\sigma_{\varepsilon}^{2}\big{)}\) with \(\varepsilon_{\varepsilon}=0.5^{\circ}\text{C}\) to account for measurement uncertainty and imperfection of the physical forward model. We generate 100 datasets each for different sample sizes \(n_{j},j=1,2,3\), with \(n_{1}=20\), \(n_{2}=100\) and \(n_{3}=500\) to assess the impact of an increasing number of observations on the performance of the method.  
**Experiment 2 (E2)**: The same test datasets as in E1 are used to address unknown measurement uncertainty.  
**Experiment 3 (E3)**: In contrast to E1, the physically realized currents are not constant, but may deviate from the set target value due to some amount of known uncertainty in the equipment used; cf. ([PERSON] et al., 2019); see Section 3.2.2, type 2a. We generate another 100 test datasets each with \(n=100\) identically manufactured LED systems as in E1, except that now we assume variability within \(I_{f,i}\), \(i=1,\ldots,n\). Thus, currents are independently sampled from a normal distribution with \(\mu_{I_{f}}=0.68\) A and \(\sigma_{I_{f}}=0.034\) A, which corresponds to a coefficient of variation of 5%. Experiment 3 strives to quantify the impact of prescribed uncertainty in \(I_{f,i}\) on \(\boldsymbol{\vartheta}_{R_{\text{th}}}\) by considering different strategies for dealing with variability within \(I_{f,i}\):  
**E3.a**: Idealized strategy: We assume that the actually realized, though random, currents are perfectly known and set to the true values \(I_{f,i}=I_{f,i}^{\text{true}}\), \(i=1,\ldots,n\) used to generate the datasets. Obviously, this first strategy is an idealized (hypothetical) and unrealistic scenario in applications. However, it will be used as a reference (benchmark) for comparison with the other strategies.
**E3.b**: Ignorant strategy: Potential variability is totally ignored within the applied currents, which are taken to be fixed at their targeted values, i.e., \(I_{f,i}=I_{f}=0.68\) A. While the data have been generated under \(I_{f,i}^{\text{true}}\), the analysis is performed under \(I_{f,i}=I_{f}\). This is probably a common scenario in practice, when variabilities in physical experiments are not considered.
**E3.c**: Proper treatment strategy: The aleatory variability in the physically realized currents is taken into account by treating \(I_{f,i}\) as an unknown parameter and assigning a prior distribution for \(I_{f,i}\) as in (7).