## 5 Results and Discussion
### Issue 1: Production and Application Variability  
Parameter estimation of \(\mathbf{\Theta}_{1}\) is performed according to the block-wise implemented RW-MH algorithm outlined in Algorithm 1 based on (16). For each dataset in E1, MCMC was run for 100,000 iterations after a burn-in phase of 20,000 draws with a thinning of 10 (i.e., 10,000 iterations are retained by keeping only every 10 th draw). We performed a number of convergence checks to assess the convergence behaviour of the Markov chain by visualizing the output using trace plots and autocorrelation function (ACF) plots of the MCMC samples; see Figure 5. Convergence to the posterior distribution is apparent from the left side of Figure 5. Additionally, the ACF of the thinned samples (after having discarded the draws of the burn-in phase) quickly decays after only a few lags. Mixing and efficiency of the MH algorithm was also evaluated on the basis of the effective sample size (ESS) and the integrated autocorrelation time (IAT); cf. [68] for details. The resulting posterior marginal distributions for \(\mu_{R_{\text{th}}}\) and \(\sigma_{R_{\text{th}}}\) for one dataset of E1 are shown in Figure 6. The estimated variability in \(R_{\text{th}}\) within a sample of \(n_{2}=100\) LED systems, which is captured by \(\sigma_{R_{\text{th}}}\), shows a good match with the true value. Table 1 summarizes the results for the test data of E1 (see 1). We report the average of the posterior means of parameters \(\mu_{R_{\text{th}}}\) and \(\sigma_{R_{\text{th}}}\), the average of the 95% highest posterior density (HPD) regions and the length of the average 95%-HPD regions over 100 datasets based on different sample sizes \(n_{1}=20\), \(n_{2}=100\) and \(n_{3}=500\). The mean and the standard deviation of the thermal resistances are well-estimated, with averaged posterior means being close to the true values that were used to generate the data. As expected, performance improves with increasing sample size. Thus, the posterior uncertainty estimated by the averaged 95%-HPD intervals steadily decreases with an increasing number of observations, which is directly evident from the HPD lengths. Providing more knowledge in the form of more data, the epistemic uncertainty in \(\boldsymbol{\vartheta}_{R_{\text{th}}}\) can be reduced by learning about \(\boldsymbol{\vartheta}_{R_{\text{th}}}\) from the data. This gain in precision is graphically displayed in Figure 7.  
Figure 5: MCMC diagnostics for \(\sigma_{R_{\text{th}}}\) for a single dataset of E1 (\(n_{2}=100\), Issue 1): (**left**) trace plot with true values (red dashed line) and burn-in phase (gray line); (**right**) ACF plot after thinning and the burn-in phase. Figure 6: Posterior marginals of (**left**) \(\mu_{R_{\text{th}}}\) and (**right**) \(\sigma_{R_{\text{th}}}\) for a single dataset of E1 (\(n_{2}=100\), Issue 1) with 95%-HPD regions, true values (red circle) and (uniform) prior distributions (yellow line). Uncertainty in the prediction of future observations is accounted for in the posterior predictive distribution \(p(\Delta T_{0_{i}}^{*}|\Delta T_{0_{i}})\), \(i=1,\ldots,n\) in (14), which is illustrated for one dataset of E1 in Figure 8 (right). The posterior predictive distribution is obtained based on samples from the marginal posterior distribution of the nuisance parameter \(R_{\text{th}}\), which is shown on the left side of Figure 8.