## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
of samples; \(N_{n}\): parameter dimensions; \(j=0,\ q_{j}=0\): initial tempering parameter \(p(\theta)\): prior distribution \(\beta\): a scaling factor of 0.2;
```  
**Algorithm 2**TMCMC [19]```
For\(i=1,2,\cdots,N\): Draw the candidate samples \(\mathbf{\theta}_{i}\)\(\sim\)\(p(\mathbf{0})\) End While\(q_{j}<1\): Transition number \(j=j+1\)  Compute tempering parameter \(q_{j}\) using Equations (18)-(20)  Compute current posterior \(p(\mathbf{0}|\mathbf{D})\) using Equation (17) based on current \(q_{j}\) For\(i=1,2,\cdots N\): Resample \(\mathbf{\theta}_{i}\) with probability \(\vec{\omega}(\mathbf{\theta}_{i})\) in Equation (21) For\(m=1,2,\cdots N_{m}\): Start MH sampling to generate new samples \(\mathbf{\theta}_{i}^{m*}\)\(\sim\)\(p(\mathbf{0}_{i}^{*}|\mathbf{\theta}_{i}^{m})\)  Proposal distribution is normal distribution with covariance \(\Sigma\) in Equation (22)  Determine if accept or reject samples \(\mathbf{\theta}_{i}^{m*}\) using Equation (23) End End  Save all samples at last stage to approximate the target posterior \(p(\mathbf{0}|\mathbf{D})\)
```

**Algorithm 2**_Cont._

### Differential Evolution Adaptive Metropolis (DREAM)

Traditional MCMC algorithms adopt a single chain or trajectory to estimate the target posterior. In contrast, DREAM simultaneously employs multiple chains in parallel to explore a global solution. DREAM was originally proposed by [PERSON] et al. (2018) and consists of two key operations, namely, differential evolution and adaptive randomized subspace sampling, leading to a strong global convergence capability. The differential evolution (DE) allows us to randomly draw sample candidates from multiple chains and enables us to efficiently determine those regions that contain global information. On the other hand, DREAM uses adaptive randomized subspace sampling to automatically tune the mean and variance of the proposal distribution, hence, not relying on artificially selected prior distribution. Building upon multiple chains, DREAM exhibits outstanding performance in estimating high-dimensional and multi-modal posteriors, as multiple chains interactively work on sampling, and the resulting collective information from these chains assists in the concurrently search for multiple local solutions ([PERSON] et al., 2018). #### 3.3.1 Differential Evolution in DREAM

The first key component of DREAM is differential evolution (DE), which controls the orientation of sample candidates toward the target posterior. Each Markov chain acts as a source of sample information for the posterior. DE fuses information from multiple chains, enhancing sample quality. Suppose we have the current sample \(\mathbf{\theta}_{i}^{c}\) that interacts with samples \(\mathbf{\theta}_{i}^{a_{1}(j)}\) and \(\mathbf{\theta}_{i}^{a_{2}(j)}\) from other chains. The proposed sample \(\mathbf{\theta}_{i}^{*}\) are generated by DE as follows:

\[\mathbf{\theta}_{i}^{*}=\mathbf{\theta}_{i}^{c}+(\mathbf{I}_{d}+\mathbf{ \Gamma}_{d})\Delta_{(\mathcal{E},d)}\left(\sum_{j=1}^{\mathcal{E}}\mathbf{ \theta}_{i}^{a_{1}(j)}-\sum_{j=1}^{\mathcal{E}}\mathbf{\theta}_{i}^{a_{2}(j) }\right)+\epsilon_{d}, \tag{24}\]

where \(i\) is the \(i\)-th Markov chain. \(\mathbf{I}_{d}\) is the identity matrix, and \(d\) is the parameter dimension. \(\mathbf{\Gamma}_{d}\) and \(\epsilon_{d}\) are perturbation values obtained from the \(d\)-dimensional multivariate uniform distribution \(U_{d}(-0.1,0.1)\) and the \(d\)-dimensional multivariate Gaussian distribution \(N_{d}(0,10^{-6})\), respectively. \(\delta\) is the number of considered chain pairs that can gradually increase the default setting of three. \(\Delta_{(\mathcal{E},d)}=2.38/\sqrt{2\delta d}\) is the incremental size of the sample change. \(\mathbf{a}_{1}(\cdot)\) and \(\mathbf{a}_{2}(\cdot)\) are two vectors containing \(\delta\) integers drawn from \(\{1,\cdots,i-1,i+1,\cdots,N\}\), \(N\) is the number of Markov chains defined in DREAM, and \(\mathbf{a}_{1}(\cdot)\
eq\mathbf{a}_{2}(\cdot)\).