## 3 Mixed-Attribute Fusion-Based Naive Bayesian Classifier
The following objective function for ANN is designed to transform one-hot encoded attributes into independent encoded attributes:  
\[\mathrm{L}\big{(}\overline{\mathrm{X}},\alpha,\beta\big{)}=\lambda\mathrm{L}_ {1}\big{(}\overline{\mathrm{X}},\alpha,\beta\big{)}+(1-\lambda)L_{2}\big{(} \overline{\mathrm{X}},\alpha,\beta\big{)}, \tag{13}\]  
where \(\lambda\in(0,1)\) is a balance factor. Optimal weight matrices \(\overline{\alpha}\) and \(\overline{\beta}\) are determined by minimizing the objective function as follows:  
\[\overline{\alpha},\overline{\beta}=\underset{ \begin{subarray}{c}\alpha_{rl}, \beta_{lr}\in\mathbb{R}\\ r=1,2,\cdots,R,3=1,2,\cdots,\mathcal{L}\end{subarray} }{\mathrm{argmax}}\big{\{} \mathrm{L}\big{(}\overline{\mathrm{X}},\alpha,\beta\big{)}\big{\}}. \tag{14}\]  
Here, it is worthwhile to note that the original intention of using an autoencoder neural network is to fuse the mixed attributes and to relax the attribute independence assumption rather than exploring the usage of deep learning technology. When a shallow learning can already meet the requirement of good NBC construction, it is unnecessary to resort to complex and time-consuming deep learning. The excessive attention on attribute transformation with deep learning is out of the scope of this study. Interested readers can refer to specialized studies on the combination of deep learning and supervised learning for more details on such approaches, e.g., deep support vector machine [22], deep decision tree [23], and deep nearest neighbor [24]. The first term of the objective function is the encoding error, which is used to measure the error between the practical output matrix \(\overline{\mathrm{X}}^{\prime}\) and the true output matrix \(\overline{\mathrm{X}}\). It is defined as follows:  
\[\mathrm{L}_{1}\big{(}\overline{\mathrm{X}},\alpha,\beta\big{)}=\left\| \overline{\mathrm{X}}^{\prime}-\overline{\mathrm{X}}\right\|_{2}^{2}=\left\| \big{[}\mathrm{sigmoid}\big{(}\overline{\mathrm{X}}\alpha\big{)}\big{]}\beta- \overline{\mathrm{X}}\right\|_{2}^{2} \tag{15}\]  
where  
\[\overline{\mathrm{X}}=\mathrm{sigmoid}\left(\overline{\mathrm{X}}\alpha\right) =\left[ \begin{array}{cccccccc}\overline{\mathrm{x}}_{1}^{(1)}&\cdots& \overline{\mathrm{x}}_{\mathcal{N}_{1}}^{(1)}\cdots&\overline{\mathrm{x}}_{1 }^{(\mathcal{M})}&\cdots&\overline{\mathrm{x}}_{\mathcal{N}_{\mathcal{M}}}^{( \mathcal{M})}\end{array} \right]^{\mathrm{T}} \tag{16}\]is a \(\mathcal{N}\times\mathcal{L}\) hidden-layer output matrix that is a class-independent encoded-attribute (IEA) data set, such that \(\overline{\Xi}_{n}^{(m)}=\left(\overline{\Xi}_{n1}^{(m)},\cdots,\overline{\Xi}_ {nC}^{(m)}\right)=\left(\text{sigmoid}\left(\overline{\Xi}_{n}^{(m)}\alpha_{1 }\right),\cdots,\text{sigmoid}\left(\overline{\Xi}_{n}^{(m)}\alpha_{\mathcal{ L}}\right)\right)\) is the independent encoded form of the original sample \(\mathbf{x}_{n}^{(m)}\) and  
\[\text{sigmoid}(s)=\frac{1}{1+\exp(-s)},s\in(-\infty,+\infty) \tag{17}\]  
is the sigmoid activation function. To minimize attribute dependence, the attribute dependence term in the objective function is defined as follows:  
\[\text{L}_{2}\left(\overline{\mathcal{X}},\alpha,\beta\right)=\frac{2}{\mathcal{ L}(\mathcal{L}-1)}\sum_{i=1}^{\mathcal{L}}\sum_{ \begin{subarray}{c}j=1\\ j\
eq i\end{subarray} }^{\mathcal{L}}\text{I}(\text{h}_{i},\text{h}_{j}), \tag{18}\]  
where  
\[\text{h}_{i}=\left(\overline{\Xi}_{1i}^{(1)},\cdots,\overline{\Xi}_{N_{1},i }^{(1)},\cdots,\overline{\Xi}_{1i}^{(\mathcal{M})},\cdots,\overline{\Xi}_{N _{M},i}^{(\mathcal{M})}\right)^{\text{T}} \tag{19}\]  
is the \(i\)-th independent encoded attribute, and \(\text{I}(\text{h}_{i},\text{h}_{j})\) is the mutual information between independent encoded attributes \(\text{h}_{i}\) and \(\text{h}_{j}\) and \(i,j\in\{1,2,\cdots,\mathcal{L}\}\) and \(i\
eq j\). The updating rules of \(\alpha_{rl}\) and \(\beta_{lr}\) are derived as follows. The second term of the objective function is unrelated to the output-layer weights. Thus, the updating rule mainly depends on the encoding error.