## 3 Methods
### 3D Prospectivity Modeling  
The MPM task aims to resolve the complicated nonlinear relationships between the predictor and mineralization variables, ultimately establishing the quantitative association model. Currently, multivariate statistical and machine learning approaches have achieved significant results in MPM [7; 11; 15; 67; 94; 95; 96]. However, the predictor variables in this study contain two distinct types: geometry-related variables and simulation-related variables. Therefore, a hierarchical modeling approach is used to combine these two types of predictor variables and understand their contributions to MPM. The Bayesian decomposition modeling (BDM) is an MPM approach that integrates nonlinearity and interpretability [97]. More importantly, the BDM method can integrate multi-type variables using the \"decomposition-integration\" strategy. Thus, the BDM was applied to MPM in this study. To integrate the two types of predictor variables, we realize MPM by two levels of BDM. At the bottom level, individual geometry- and simulation-related predictor variables are integrated to generate their distinctive feature representations. At the top level, we integrate geometry- and simulation-related feature representations and map them to ore-bearing probability. Firstly, assuming that the geometry- and simulation-related variable sets are denoted as \(E_{g}\) and \(E_{s}\), respectively, the mineralization probability based on the Bayes' rule can be expressed as:  
\[P(Ore=1|E_{g},E_{s})=\frac{P\big{(}E_{g},E_{s}|Ore=1\big{)}P(Ore=1)}{P\big{(}E _{g},E_{s}\big{)}}. \tag{1}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Property** & \
\begin{tabular}{l} **Hanging Wall** \\ **(Metamorphic Rocks)** \\ \end{tabular} &
\begin{tabular}{l} **Footwall** \\ **(Granite)** \\ \end{tabular} & **Fault Zone** \\ \hline Cohesion (Pa) & 8.9 & 33.6 & 16.9 \\ Tensile strength (Pa) & 3.77 & 5.16 & 2.97 \\ Friction angle (\({}^{\circ}\)) & 28 & 33 & 20 \\ Dilation angle (\({}^{\circ}\)) & 3.2 & 5.3 & 6.0 \\ Porosity & 0.29 & 0.19 & 0.4 \\ Permeability (m\({}^{2}\)) & 2.09 \(\times\) 10\({}^{-11}\) & 1.81 \(\times\) 10\({}^{-11}\) & 1.00 \(\times\) 10\({}^{-10}\) \\ Conductivity (J/(m\({}^{\circ}\)C)) & 2.63 & 3.05 & 4.0 \\ Thermal expansion & 5.40 \(\times\) 10\({}^{-6}\) & 6.70 \(\times\) 10\({}^{-6}\) & 6.70 \(\times\) 10\({}^{-6}\) \\ coefficient (\({}^{\circ}\)C\({}^{-1}\)) & 5.40 \(\times\) 10\({}^{-6}\) & 6.70 \(\times\) 10\({}^{-6}\) \\ Specific heat & & & \\ capacity (J/(kg\({}^{\circ}\)C)) & 803 & 803 & 783 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._According to the derivation in [97], Equation (1) can be formulated as:  
\[P(Ore=1|E_{g},E_{s})=\frac{1}{1+\exp\bigl{(}-\bigl{(}\theta_{g}\phi_{g}\bigl{(}E_{ g}\bigr{)}\!+\!\theta_{s}\phi_{s}(E_{s})+b\bigr{)}\bigr{)}}, \tag{2}\]  
where \(\theta_{g}\) and \(\theta_{s}\) are the relevant indices that express the correlation among the variable sets \(E_{g}\) and \(E_{s}\), while \(\phi_{g}\bigl{(}E_{g}\bigr{)}\), \(\phi_{s}(E_{s})\) and \(b\) represent the nonlinear mapping functions (NMFs) and the prior correction, respectively:  
\[\left\{ \begin{array}{l}\phi_{g}\bigl{(}E_{g}\bigr{)}=\log\frac{P\bigl{(}Ore= 1|E_{g}\bigr{)}}{P\bigl{(}Ore=0|E_{g}\bigr{)}}=\log\frac{P\bigl{(}Ore=1|E_{g} \bigr{)}}{1-P\bigl{(}Ore=1|E_{g}\bigr{)}}\\ \phi_{s}(E_{s})=\log\frac{P\bigl{(}Ore=1|E_{s}\bigr{)}}{P\bigl{(}Ore=0|E_{s} \bigr{)}}=\log\frac{P\bigl{(}Ore=1|E_{s}\bigr{)}}{1-P\bigl{(}Ore=1|E_{s} \bigr{)}}\\ b=\bigl{(}1-\theta_{g}-\theta_{s}\bigr{)}\log\frac{P\bigl{(}Ore=1\bigr{)}}{P \bigl{(}Ore=0\bigr{)}}\end{array}