## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
DREAM defines \(\Delta_{(\delta,\delta)}\) as one of every five chains that jumps from one posterior to another posterior, resulting in \(p_{\Delta_{(\delta,\delta)}}=0.2\). #### 3.3.2 Adaptive Randomized Subspace Sampling

In high-dimensional parameter spaces, optimal samples may not be drawn in full dimensions [50]. Adaptive randomized subspace sampling iteratively modifies the proposal variations and updates parameter dimensions based on historical samples. A new subspace is formed by selecting a d-dimensional subset of the parameter space. A crossover probability (CR) determines whether to modify each parameter dimension

\[\theta_{i}^{t,j}=\left\{ \begin{matrix}\theta_{i}^{t-1,j},u<1-\hat{\alpha},\\ \theta_{i}^{p,j},\end{matrix} \right.\quad\text{\emph{else}}, \tag{25}\]

where \(\theta_{i}^{t-1,j}\) is the \(d\)-th dimensional sample of the \(i\)-th chain at the \((t-1)\)-th iteration. \(u\) is a value sampled from a uniform distribution \(U(0,1)\). \(\hat{\alpha}\) is a crossover value that is sampled from a discrete multi-nominal distribution, that is, \(\hat{\alpha}\sim p(\mathbf{CR},\mathbf{f}_{\mathbf{\text{CR}}})\), where \(\mathbf{CR}=\left\{\frac{1}{n_{cr}},\frac{2}{n_{cr}},\cdots,1\right\}\) with a different CR for each element in the \(\mathbf{CR}\) vector. \(n_{cr}\) is the number of CR values and is user-specified. The recommendation of \(n_{cr}\) is three by [PERSON] et al. [50]. The interpretation of Equation (25) is that the current sample is retained if \(u<1-\hat{\alpha}\); otherwise, it is updated. Note that the adaptive randomized subspace sampling strategy is activated when a crossover value \(\hat{\alpha}<1\), and a vector of sample candidates \(\mathbf{\uptheta}_{i}^{p,j}=\left\{\theta_{i}^{p,1},\theta_{i}^{p,2},\cdots, \theta_{i}^{p,j}\right\}\) is then constructed from a multivariate normal distribution. DREAM adaptively tunes the crossover probability by optimizing the distance between two adjacent chains, resulting in a decrease in autocorrelation among the samples in each chain. In particular, when CR = 1, all dimensions are updated; when CR = 0, a single dimension that is randomly selected from \(d\) dimensions to avoid the zero value of jump rate \(\Delta_{(\delta,d)}\) is updated. #### 3.3.3 Detection and Handling of Aberrant Chains

DREAM can detect aberrant chains among multiple chains. Large discrepancies between initial and posterior samples often occur when initial chains use a high-variance prior, slowing convergence. Aberrant chains, stuck in local optima, impair distribution estimates and slow evolution. DREAM uses the inter-quartile range (IQR) method to track abnormal chain status [50]

\[\text{IQR}=Q_{3}-Q_{1}, \tag{26}\]

where \(Q_{1}\) and \(Q_{3}\) are, respectively, the lower and upper quartiles of the Markov chain computed based on samples before the current time. Chains with \(\mathcal{H}<Q_{1}-2\cdot\)IQR are detected as aberrant, and \(\mathcal{H}\) is the mean of the logarithm of the posterior distribution of the last half samples in each chain. Equation (26) serves to ensure each remaining Markov chain is stable from a statistical perspective. #### 3.3.4 Convergence Evaluation

DREAM evaluates the convergence quality of multiple chains using Gelman-Rubin statistics also known as scale reduction factor \(R_{stat}\)[51], as a convergence criterion to determine whether the sampling terminates or not. DREAM algorithm stipulates that if \(R_{stat}<1.2\) for all unknown parameters, a stable posterior PDF is achieved. \(R_{stat}\) is expressed as follows [50]:

\[R_{stat}=\sqrt{\frac{\gamma-1}{\gamma}+\frac{N+1}{N\cdot Z}\frac{B}{\gamma}}, \tag{27}\]where \(\gamma\) is the number of iteration samples of each chain; \(N\) is the number of Markov chains used for sampling; \(Z\) is the mean of the variance of total \(N\) Markov chains; and the ratio of \(B/\gamma\) is the variance of the mean of \(N\) parallel Markov chains. The implementation description of the DREAM algorithm is presented in Algorithm 3. ```
\(\boldsymbol{\#}\) Initialization \(N\): the No.