## 5 Results and Discussion
### Issue 3: Uncertain Experimental Conditions  
The setup in E3 aims to compare different strategies for dealing with uncertain experimental conditions. As with the previous issues, MCMC was run for 100,000 iterations with a burn-in of 20,000 iterations and a thinning of 10 for strategies E3.a and E3.b to estimate the parameters \(\mu_{R_{\mathrm{th}}}\) and \(\sigma_{R_{\mathrm{th}}}\). For each dataset, the modified MCMC algorithm in E3.c was run for 200,000 iterations after a burn-in of 100,000 draws with a thinning of 10 to ensure convergence to the target distribution. Results are summarized in Table 2, where the averaged posterior means for \(\mathbf{\vartheta}_{R_{\mathrm{th}}}\) under the three strategies and the averaged lower and upper bounds of 95%-HPD regions and the respective averaged 95%-HPD region lengths over 100 datasets of size \(n=100\) are reported.  
Figure 10 illustrates the distribution of the averaged posterior means for \(\mathbf{\vartheta}_{R_{\mathrm{th}}}\) over the simulated datasets for the three strategies, where the first (idealized) one serves as the reference (\"what if\") strategy. Averages of the posterior means for \(\mu_{R_{\mathrm{th}}}\) agree well with the  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Issue**} & \multirow{2}{*}{**Parameter**} & **True** & **Strategy** & **Av. Post.** & **Av. 95\%-** & **Length** \\  & & **Value** & & **Mean** & **HPD** & **HPD** \\ \hline (I3) Uncertain experimental conditions & \(\mu_{R_{\mathrm{th}}}\) & 39.13 & 1. idealized & 39.130 & [38.930,39.332] & 0.402 \\  & & & 2. ignorant & 39.134 & [38.696,39.574] & 0.878 \\  & & & 3. proper & 39.134 & [38.745,39.541] & 0.796 \\ \cline{3-6}  & \(\sigma_{R_{\mathrm{th}}}\) & 1.00 & 1. idealized & 1.011 & [0.869,1.162] & 0.293 \\  & & & 2. ignorant & 2.231 & [1.924,2.558] & 0.633 \\  & & & 3. proper & 1.051 & [0.614,1.502] & 0.888 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation results for Issue 3 (I3, Section 4.4) based on experiments E3.a–E3.c. Averaged posterior means (Av. post. mean) for \(\mu_{R_{\mathrm{th}}}\) and \(\sigma_{R_{\mathrm{th}}}\), averaged lower and upper bounds of 95%-HPD regions (Av. 95%-HPD) for \(\mu_{R_{\mathrm{th}}}\) and \(\sigma_{R_{\mathrm{th}}}\), and respective length of the averaged 95%-HPD regions (Length HPD) over 100 datasets based on \(n=100\) observations.  
Figure 9: Posterior distribution of \(\sigma_{\mathrm{\varepsilon}}\) for a single dataset of E2 (Issue 2) with 95%-HPD regions, true value (red circle) and prior distribution \(p(\sigma_{\mathrm{\varepsilon}})\sim\mathcal{B}(a_{0},b_{0})\) (yellow line). (**Left**) \(n_{2}=100\) and \(a_{0}=b_{0}=12\); (**right**) \(n_{3}=500\) and \(a_{0}=b_{0}=40\).  
true values under all strategies, but E3.b (ignorant) and E3.c (proper) show considerably more variation than E3.a (idealized). This is also evident from the respective length of the averaged 95%-HPD regions.  
However, for the variance parameter \(\sigma_{\text{R}_{\text{th}}}\), we observe a different result. If aleatory variability in the experimental setup is not taken into account and the currents are erroneously taken on their targeted values (E3.b), we get misleading estimates of the posterior mean for \(\sigma_{\text{R}_{\text{th}}}\). Thus, the additional variability in the observed data induced by the equipment used is apparently attributed to \(\sigma_{\text{R}_{\text{th}}}\), over-estimating the variability within the thermal resistances. Consequently, inference of the predictive distribution for \(\Delta T_{0_{i}}\) will be biased. In contrast, if the variability in the currents is properly considered by treating them as uncertain (E3.c), we observe that the posterior means are much closer to the true value, but the additional variability in the currents has increased the level of estimation uncertainty.