## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
One solution for this shortcoming is to determine better initial samples using global optimization methods, e.g., heuristic optimization [80]. TMCMC generates samples from a series of transitional distributions and has better capabilities for effectively sampling from high-dimensional posteriors compared to DRAM. For moderately high dimensions, e.g., 15 dimensions in Section 4.3, the accuracy of damage detection is acceptable. Nevertheless, for a higher dimensional problem, e.g., 40 dimensions in Section 4.1, its accuracy substantially degrades, even providing misguiding results for damage detection. Among the three sampling algorithms, TMCMC has the fastest convergence speed. In other words, the dropping burn-in period is not an issue in TMCMC. This is attributed to the fact that the initial sample pool in the TMCMC algorithm is directly built based on the prior distribution of model parameters, which bypasses sampling from outside the posteriors. However, the burn-in period may occur in some situations in which the target posterior is very complex and high-dimensional or is distributed over a relatively narrower region [34]. The disadvantages of TMCMC mainly lie in algorithm complexity. The computational cost required in the entire sampling process is more expensive than the process using DRAM. TMCMC does not generate samples from the target posteriors but from intermediate distributions, which means that more parameters at each iteration, \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multicolumn{3}{c}{**Accuracy**} & \multicolumn{3}{c}{**Computational Cost (min)**} & \multirow{2}{*}{**Complexity**} \\ \cline{2-2} \cline{6-7}  & **DRAM** & **TMCMC** & **DREAM** & **DRAM** & **TMCMC** & **DREAM** & \\ \hline Shear building & Low & Medium & High & 15 & 8 & 40 & Single-chain \\ Continuous beam & Low & Medium & High & 10 & 14 & 50 & Single-chain \\ Pedestrian bridge & Low & Medium & High & 4 & 25 & 30 & Multiple-chain \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison among DRAM, TMCMC, and DREAM. e.g., mean and covariance matrix in Equation (22), need to be iteratively tuned. Another problem in TMCMC is the selection of the scaling parameter \(\beta\), which is not universal, although it has been suggested as 0.2 in [19]. However, a value of 0.2 may not be optimal and applicable for all cases. A remedial option is to adaptively tune \(\beta\) based on the comparison between the mean acceptance rate and target acceptance rate [34]. DREAM employed multiple Markov chains in parallel and differential evolutions as well as randomized subspace strategies in order to sample the posterior distributions. As observed in all demonstration examples, DREAM exhibits the most accurate and reliable performance of damage detection among all algorithms. DREAM allows the sharing of information from each Markov chain. The information fusion by DREAM's collective power significantly enhances the convergence accuracy and quality of the posterior samples, leading to the best ability in estimating parameters for high-dimensional problems. The main input required to implement DREAM is a number of Markov chains. Other inputs have been elaborately investigated and are recommended by [PERSON] et al. [50]. It was found that default algorithm settings work very well for the presented engineering examples. Another key strength is that the issue of initial values is less of a concern in DREAM, as the initial samples are generated using Latin hypercube samples with a physical parameter bound. However, DREAM's outstanding capability for approximating complex and high-dimensional posteriors comes at the expense of needing to generate samples from different Markov chains. The sample size is much larger than in DRAM and TMCMC, undoubtedly increasing the computational time for damage detection. The time required by DREAM was consistently the highest among the three sampling algorithms. To alleviate the computational burden, one can turn to the parallel computing of Markov chains.