# Dempster-Shafer Fusion Based on a Deep Boltzmann Machine for Blood Pressure Estimation
## 3 DBM-Based DS Fusion
### Dbm  
DBM proposed in [13] was determined to improve the inference during the DBN learning process. As shown in Figure 3, we used a two-hidden-layer DBM to reduce complexity. The energy of the state, \([\mathbf{x},\mathbf{h}^{1},\mathbf{h}^{2}]\), is defined as:  
\[\mathbb{E}(\mathbf{x},\mathbf{h}^{1},\mathbf{h}^{2};\theta)=-\mathbf{x}^{ \mathrm{T}}W^{1}\mathbf{h}^{1}-\mathbf{h}^{\mathrm{T}}W^{2}\mathbf{h}^{2}, \tag{1}\]  
where \(\theta=[W^{1},W^{2}]\) are parameters that express the symmetric interaction terms of the input units to hidden units, and the hidden units to hidden units [13]. In addition, \(i\), \(j\), and \(m\) denote the indexes of the input unit, hidden unit \(\mathbf{h}^{1}\), and hidden unit \(\mathbf{h}^{2}\), respectively. The probability is assigned to an input data, \(\mathbf{x}\), defined by:  
\[p(\mathbf{x};\theta)=\frac{\sum_{\mathbf{h}^{1},\mathbf{h}^{2}}\exp(-\mathbb{ E}(\mathbf{x},\mathbf{h}^{1},\mathbf{h}^{2};\theta))}{\sum_{\mathbf{x}}\sum_{ \mathbf{h}^{1},\mathbf{h}^{2}}\exp(-\mathbb{E}(\mathbf{x},\mathbf{h}^{1}, \mathbf{h}^{2};\theta))}. \tag{2}\]  
The distributions over the input vector and the two sets of hidden units are expressed by:  
\[p(h^{1}_{j}=1|\mathbf{x},\mathbf{h}^{2})=\phi\bigg{(}\sum_{i}W^{1}_{ij}x_{i}+ \sum_{m}W^{2}_{jm}h^{2}_{j}\bigg{)}, \tag{3}\]  
\[p(h^{2}_{m}=1|\mathbf{h}^{1})=\phi\bigg{(}\sum_{j}W^{2}_{im}h^{1}_{i}\bigg{)}, \tag{4}\]  
\[p(x_{i}=1|\mathbf{h}^{1})=\phi\bigg{(}\sum_{j}W^{1}_{ij}h^{1}_{j}\bigg{)}, \tag{5}\]  
Figure 2: The CDF (cumulative distribution function) plots of artificial features (**a**) MAP (mean arterial pressure); (**b**) AR; (**c**) AE; (**d**) LE; (**e**) MA; (**f**) \(\sigma 1\); (**g**) \(\sigma 2\); (**h**) LMAP; (**i**) AR2; (**j**) AR3; (**k**) AE2; (**l**) AE3; (**m**) gender; (**n**) HR; and the subject’s age are investigated as the features and generated using the bootstrap technique with replication numbers (\(B=100\)) from one volunteer [5,12]. where \(\phi(*)=1/(1+\exp(-*)\). The generative probability is defined by:  
\[p(\mathbf{x};\theta)=\sum_{\mathbf{h}^{1}}p(\mathbf{h}^{1}_{j};\mathbf{W}^{1})p( \mathbf{x}|\mathbf{h}^{1};\mathbf{W}^{1}), \tag{6}\]  
where \(p(\mathbf{h}^{1};\mathbf{W}^{1})=\sum_{\mathbf{x}}P(\mathbf{x}|\mathbf{h}^{1} ;\mathbf{W}^{1})\) denotes an implicit prior probability over \(\mathbf{h}^{1}\) given by the parameters. The second RBM in the stack substitutes \(p(\mathbf{h}^{1};\mathbf{W}^{1})\) by \(p(\mathbf{h}^{1};\mathbf{W}^{2})=\sum_{\mathbf{h}^{2}}p(\mathbf{h}^{1}, \mathbf{h}^{2};\mathbf{W}^{2})\). When the second RBM is correctly initialized, \(p(\mathbf{h}^{1};\mathbf{W}^{2})\) is a better probability of all posterior distributions of \(\mathbf{h}^{1}\), where the all posterior is the non-factorial mixture of the factorial posteriors as \(\frac{1}{N}\sum_{n}p(\mathbf{h}^{1}|\mathbf{x}_{n};\mathbf{W}^{1})\). Here, \(N\) denotes the size of the input vector. Note that it is possible to infer \(P(\mathbf{h}^{1};\mathbf{W}^{1},\mathbf{W}^{2})\) through averaging the two models of \(\mathbf{h}^{1}\), which is approximately performed by utilizing the \(1/2\mathbf{W}^{1}\) bottom-up approach and \(1/2\mathbf{W}^{2}\) top-down approach because the second RBM replaces \(p(\mathbf{h}^{1};\mathbf{W}^{1})\) with a better model [13]. The pre-training is presented [13] to learn a stack of RBMs.