## 3 Model and Methods
### Bayesian Hierarchical Modelling Framework
#### 3.2.3 Bayesian Hierarchical Model  
Having specified the likelihood distribution and the priors, the overall Bayesian hierarchical (or multi-level) model is summarized in the following way:  
\[y_{i}|R_{\text{th}_{i}},I_{f,i},\mathbf{z}_{i} \sim p(y_{i}-\mathcal{M}(R_{\text{th}_{i}},I_{f,i},\mathbf{z}_{i})|\sigma _{z}^{2}) \tag{11}\] \[R_{\text{th}_{i}} \sim p(R_{\text{th}_{i}}|\mathbf{\vartheta}_{R_{\text{th}}})\] \[\mathbf{\vartheta}_{R_{\text{th}_{i}}} \sim p(\mathbf{\vartheta}_{R_{\text{th}}}|\mathbf{\gamma}_{R_{\text{th}}})\] \[I_{f,i} \sim p(I_{f,i}|\mathbf{\vartheta}_{I_{f,i}})\]  
with \(\mathbf{z}_{i}=(\eta_{h},V_{f})\), and it can be represented as a probabilistic graphical model (PGM), cf. [55; 56; 57]), or, more specifically, as a directed acyclic graph (DAG), also known as Bayesian network (BN) [58]. The joint posterior distribution of the unknown parameters \(\mathbf{\Theta}=(\mathbf{R}_{\text{th}},\mathbf{\vartheta}_{R_{\text{th}}},\mathbf{I}_{f})\) is obtained by combining the likelihood in (6) and the joint prior (10) into the posterior distribution, i.e.,  
\[p(\mathbf{R}_{\text{th}},\mathbf{\vartheta}_{R_{\text{th}}},\mathbf{I}_{f}| \mathbf{y}) = \frac{1}{C}\prod_{i=1}^{n}p(y_{i}|R_{\text{th}_{i}},I_{f_{i}},\bm {z}_{i})\cdot\left(\prod_{i=1}^{n}p(I_{f,i}|\mathbf{\theta}_{I_{f,i}})\right)\cdot\] \[\left(\prod_{i=1}^{n}p(R_{\text{th}_{i}}|\mathbf{\vartheta}_{R_{ \text{th}}})\right)\cdot p(\mathbf{\vartheta}_{R_{\text{th}}}).\]  
The normalizing constant \(C\) of the posterior is given as  
\[C=\int_{I_{f}^{n}}\int_{R_{\text{th}}^{n}}\int_{\theta_{R_{\text{th}}}}p(\mathbf{ R}_{\text{th}},\mathbf{\vartheta}_{R_{\text{th}}},\mathbf{I}_{f}|\mathbf{y})\ d\mathbf{I}_{f}\ d\mathbf{R}_{\text{th}}\ d\mathbf{\vartheta}_{R_{\text{th}}}.\]  
Based on (12), inference on specific parameters of interest can be achieved by marginalization. The corresponding DAG representation of the joint posterior is shown in Figure 4.  
It provides a model representation for the hierarchical structure of the joint probability distribution over a set of random variables in terms of prior and conditional distributions,  
Figure 4: Representation of the BHM in (11) as a directed acyclic graph (DAG).  
linking parameters, hyper-parameters and the observed data by using concepts of graph theory and conditional independence among random variables; see [24; 59].  
The posterior distribution in (12) accounts for random uncertainty in the generated data, induced, e.g., by measurement noise, through the likelihood as well as for epistemic parameter uncertainty via the prior. Thus, exploring the posterior distribution opens up a natural framework for UQ by computing credible intervals and posterior predictive distributions. Posterior uncertainty related to \(\mathbf{\Theta}\) can be quantified using credible regions based on quantiles of the posterior distribution; cf. [22]. Uncertainty in the prediction of future observations \(\mathbf{y}^{*}\) is fully considered within the Bayesian framework by providing the posterior predictive distribution \(p(\mathbf{y}^{*}|\mathbf{y})\), which quantifies the predictive uncertainty in \(\mathbf{y}=\Delta\mathbf{T}_{0}\) and is given as  
\[p(\mathbf{y}^{*}|\mathbf{y})=\int_{\Theta}p(\mathbf{y}^{*}|\mathbf{\Theta},\mathbf{z})\cdot p(\bm {\Theta}|\mathbf{y},\mathbf{z})\;d\mathbf{\Theta}. \tag{13}\]  
See, e.g., [23; 60] for details. Hence, the posterior predictive distribution reflects two types of uncertainty: the uncertainty in sampling future data (sampling uncertainty) through \(p(\mathbf{y}^{*}|\mathbf{\Theta},\mathbf{z})\), and parametric (estimation) uncertainty about \(\mathbf{\Theta}\) through the posterior distribution, i.e., \(p(\mathbf{\Theta}|\mathbf{y},\mathbf{z})\).