## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.2 Mathematical Formulation  
TMCMC constructs a series of simple transitional distributions starting from the prior distribution and converging on the target distribution, which is mathematically expressed as  
\[p(\theta|\mathbf{D})\propto p(\mathbf{D}|\theta)^{q_{j}}\cdot p(\theta)\ (j=0,1,\cdots,m;0=q_{0}<q_{1}<\cdots<q_{m}=1), \tag{17}\]  
where \(j\) is the transition stage number and \(m\) is the total number of transition stages. \(p(\mathbf{D}|\theta)\) is the intermediate distribution controlled by tempering parameter \(q_{j}\) that takes values from zero and one. At \(q_{0}=0\), the initial stage with zero tempering parameters, the posterior \(p(\theta|\mathbf{D})\) is proportional to the prior \(p(\theta)\). At \(q_{m}=1\), the final stage with the unity tempering parameter, the intermediate distribution converges on the target distribution. By progressively increasing \(q_{j}\) toward unity, TMCMC calibrates the intermediate distribution from the prior to the target distribution.  
The critical part in TMCMC is the determination of the tempering parameter \(q_{j}\) that controls the shape variation between two successive intermediate distributions. \(q_{j}\) should be chosen so that its variation, \(q_{j}-q_{j-1}\), allows the smooth transition from \(p(\mathbf{D}|\theta)^{q_{j-1}}\) to \(p(\mathbf{D}|\theta)^{j}\). The acceptance rate is greatly affected by the selection of \(q_{j}\). Large variations cause sharp changes in the intermediate distribution, leading to high rejection rates with many abandoned samples. Conversely, extremely small variations imply inefficacious sampling with a 100% acceptance rate as all samples are generated from the same distribution [34]. Therefore, an optimal strategy for \(q_{j}\) is required. Once \(q_{j}\) is determined, samples at stage \(j\) can be generated from the previous distribution \(p(\mathbf{D}|\theta)^{q_{j-1}}\cdot p(\theta)\). [PERSON] and [PERSON] [19] recommend using the coefficient of variation (COV) of the so-called plausibility weight of the sample \(\theta_{i}\) to determine the optimal \(q_{j}\), which is defined as  
\[\omega^{j}=p(\mathbf{D}|\theta_{i})^{q_{j}-q_{j-1}}. \tag{18}\]  
Based on the definition of COV, the COV of plausibility weight is given by  
\[\text{COV}\big{(}q_{j}\big{)}=\frac{\sigma\big{(}\omega^{j}\big{)}}{\mu\big{(} \omega^{j}\big{)}}, \tag{19}\]  
where \(\sigma(\cdot)\) and \(\mu(\cdot)\) are the standard deviation and the mean of \(p(\mathbf{D}|\theta_{i})^{q_{j}-q_{j-1}}\), where \(\theta_{i}=\big{(}\theta_{i}^{1},\theta_{i}^{2},\cdots,\theta_{i}^{N_{m}} \big{)}\) and \(i=1,2,\cdots,N\) (in which \(N\) is the number of samples and \(N_{m}\) is parameter dimension). \(\text{COV}\big{(}q_{j}\big{)}\) in Equation (19) is a function of the \(q_{j}\) conditional on a known \(q_{j-1}\). One could prescribe a threshold \(\epsilon\) to satisfy the requirement \(\text{COV}\big{(}q_{j}\big{)}\leq\epsilon\); thus, the optimal \(q_{j}\) for each iteration can be automatically computed. Another way to determine \(q_{j}\) is to minimize the absolute difference between \(\text{COV}\big{(}q_{j}\big{)}\) and unity in order to preserve COV as the most unified approach to possible. Thus, we obtain  
\[q_{j}=\underset{q_{j}}{\text{argmin}}\big{(}|\text{COV}\big{(}q_{j}\big{)}-1 |\big{)}. \tag{20}\]