## 2 Methodology
### Hamiltonian Monte Carlo (HMC) Sampling Method  
The choice of approximation methods and samplers for of Bayesian parameter estimation depends on the characteristics of posterior distribution. For instance, the commonly used MCMC algorithms, such as random walk Metropolis and Gibbs sampling, are morechallenging to tune and converge [28] and less efficient in generating large effective sample sizes [29] for high-dimensional distribution functions, e.g., the improved Weibull model in this study, in comparison with the HMC method and the No-U-turn Sampler (NUTS) that extends HMC. HMC is a gradient sampling method established from the Hamiltonian Dynamics theory, rather than the random-walk mechanism, to predict sample locations, thus achieving a higher sampling rate.  
Hamiltonian dynamics describe the energy transfer between kinetic energy and potential energy of particles moving in a system. Assuming a particle's position vector is \(s\) and its velocity vector is \(\phi\), the state of the particle can be described as \(\chi=(s,\phi)\), and the joint canonical distribution of the two vectors is:  
\[\pi(s,\phi)=\pi(\phi|s)\pi(s) \tag{18}\]  
Assume that the _Hamiltonian_, \(H\)(\(s\),\(\phi\)), satisfies:  
\[\pi(s,\phi)\propto\exp[-H(s,\phi)] \tag{19}\]  
where:  
\[H(s,\phi)=E(s)+K(\phi)=E(s)+\frac{1}{2}{\sum_{i}{\phi_{i}}^{2}} \tag{20}\]  
where \(E\)(\(s\)) and \(K\)(\(\phi\)) denote potential energy and kinetic energy, and are defined as follows:  
\[\begin{array}{l}K(\phi)=-\log\pi(\phi)\\ E(s)=-\log\pi(s)\end{array} \tag{21}\]  
If \(\pi\)(\(s\)) is the target distribution to sample, the HMC method does not directly generate a sample of \(\pi\)(\(s\)). Rather, it simulates a sample from the joint canonical distribution \(\pi\)(\(s\),\(\phi\)), which can be specified by putting Equations (20) and (21) into Equation (19) as:  
\[\pi(s,\phi)=\frac{1}{Z}\exp[-H(s,\phi)]=\pi(s)\pi(\phi) \tag{22}\]  
where \(Z\) is a normalizing constant, and \(\pi\)(\(s\)) and \(\pi\)(\(\phi\)) are independent from each other. The independency between \(\pi\)(\(s\)) and \(\pi\)(\(\phi\)) means that the sampling of \(s\) via the sample of \(\pi\)(\(s\),\(\phi\)) with any predetermined \(\pi\)(\(\phi\)) can well fit the distribution of \(\pi\)(\(s\)). In this sampling process, \(\pi\)(\(\phi\)) acts as an auxiliary variable, generally being designated as normal distribution \(N\) (0,1).  
According to the Hamiltonian equations of the Hamiltonian Dynamics:  
\[\begin{array}{l}\frac{ds_{i}}{dt}=\frac{\partial H}{\partial\phi_{i}}=\phi_ {i}\\ \frac{d\phi_{i}}{dt}=-\frac{\partial H}{\partial s_{i}}=-\frac{\partial E}{ \partial s_{i}}\end{array} \tag{23}\]  
the values of \(s\) can be derived just by knowing \(\partial E/\partial s_{i}\).  
In practice, the aforementioned sampling process needs to be simulated by a leap-frog iterative algorithm, which includes three steps:  
\[\begin{array}{l}\phi_{i}(t+\varepsilon/2)=\phi_{i}(t)-\frac{\varepsilon}{2 }\frac{\partial}{\partial s_{i}}E[s(t)]\\ s_{i}(t+\varepsilon)=s_{i}(t)+\varepsilon\phi_{i}(t+\varepsilon/2)\\ \phi_{i}(t+\varepsilon)=\phi_{i}(t+\varepsilon/2)-\frac{\varepsilon}{2}\frac{ \partial}{\partial s_{i}}E[s(t+\varepsilon/2)]\end{array} \tag{24}\]  
Equation (24) accomplishes the state transition via a change in the velocity variable \(\phi\) with half a time step, while ensuring that the state transition process is reversible and the _Hamiltonian_\(H\)(\(s\),\(\phi\)) remains unchanged in this process. Subsequently, integrated with the aid of NUTS, the HMC sampling process can achieve a much higher sampling rate than other purely random sampling methods.