## 5 Discussion  
### Convergence Observations--Number of Simulations  
To determine how many simulations are required for numerical stability, the convergence of the mean and standard deviation of FS are examined for both baseline and weighted analyses. They are evaluated by performing 100 additional complete Monte Carlo analyses with logarithmically spaced sample sizes from 1 to \(10^{6}\) and plotting \(\upmu\) and \(\sigma\) versus simulation count. Additionally, the KL (FS) convergence curve is created within the same numerical procedure. Figure 18 displays the convergence curves for the mean and standard deviation of FS distribution. It indicates that 10,000 simulations are optimal: beyond this, \(\upmu\) and \(\sigma\) of the weighted analysis histogram no longer change appreciably. The baseline and weighted analysis do not converge with the same number of simulations. The computational cost of the weighted analysis depends on the _effective_ subset of the nominal number of simulations that retain non-negligible weight after the monitoring data filter has been applied. This is referred to as the weighted simulation count (sum of all weights) \(N^{weighted}\). Figure 18 shows that the baseline Monte Carlo results stabilize with roughly 300 simulations, whereas the weighted analysis attains convergence only when the nominal simulation count is increased to about \(10^{4}\). The disparity arises because, after weighting, those \(10^{4}\) simulations contribute an effective count of just 235 (Figure 17), which is comparable to the 200 simulations that suffice for the unweighted case. Figure 19 demonstrates that the KL divergence exhibits considerable variability at small sample sizes due to the insufficient population of the weighted histograms. Once the simulation count surpasses approximately \(10^{4}\) simulations, the KL converges to a steady value of approx. 1.6 nats--a plateau that matches the reported KL value in Figure 17. Beyond this threshold, additional simulations yield negligible further change of KL divergence, signifying that the informational content conveyed by the monitoring data has been effectively exhausted. Figure 18: Convergence curves for (**a**) mean of FS; (**b**) standard deviation of FS. The weighted simulation count \(N^{weighted}\) is presented in Figure 20 against the standard deviation of the monitoring data (\(\sigma_{mnn}\)), for four nominal simulation counts \(N\). Four Monte Carlo sets of samples are created: a barely converging set of 200 simulations and three progressively larger sets of \(10^{3}\), \(10^{4}\) and \(10^{5}\) samples. For each set, the complete weighted analysis is repeated 100 times while the monitoring standard deviation \(\sigma_{mnn}\) is varied logarithmically from 0.0002 to 0.2 (0.1-99% \(\epsilon_{1,\text{range}}\) from baseline MC analysis). For each \(\sigma_{mnn}\) value, the weights are summed up to obtain the \(N^{weighted}\). The whole described procedure is then repeated 50 times with different \(N\) samples in each of the four sets. This produces 50 values of the \(N^{weighted}\) for each \(\sigma_{mnn}\). Figure 20 plots the mean of those 50 values as the solid curves and shades the regions covering \(\pm 2\) standard deviations of \(N^{weighted}\) distributions. The shaded envelopes therefore visualize the spread of \(N^{weighted}\) that arises purely from the random selection of nominal simulations \(N\). When the monitoring standard deviation is roughly one percent (0.002) or more of the baseline spread, a nominal set of about \(10^{4}\) Monte Carlo simulations already supplies several hundred effective draws, yielding a well-resolved posterior at a modest cost. If the monitoring span tightens to a few-tenths of a percent, however, the effective sample size collapses below the practical threshold of 200, and the posterior becomes noisy unless the nominal simulation count \(N\) is expanded by several orders of magnitude. The shaded \(\pm 2\)\(\sigma\)  
Figure 19: Convergence curve for KL (FS).