## 3 Switched Variational Estimation Filtering in Underwater Navigation  
### Mixture Probabilistic Model  
According to statistic theory, \(\xi(t_{k})\) is depended on \(\tau(t_{k})\); then, the expression of \(\text{Pro}[\xi(t_{k})|\tau(t_{k})\,]\) is:  
\[\text{Pro}[\xi(t_{k})|\tau(t_{k})\,]=[\tau(t_{k})]^{\xi(t_{k})}[1-\tau(t_{k}) ]^{[1-\xi(t_{k})]} \tag{5}\]  
\(\mathbf{Z}(t_{k}  
**Assumption 1**. : _The PDF of \(\mathbf{Z}(t_{k})\) is assumed to follow the Gaussian distribution. Therefore, \(\mathrm{Pro}[\mathbf{Z}(t_{k})|\mathbf{X}(t_{k}),\xi(t_{k})]\) is expanded as:_  
\[\mathrm{Pro}[\mathbf{Z}(t_{k})|\mathbf{X}(t_{k}),\xi(t_{k})\,]=N^{\xi(t_{k})} \big{(}\mathbf{Z}(t_{k});h[\mathbf{X}(t_{k})],\mathbf{P}\big{(}t_{k}^{+}\big{)} \big{)}\ \times N^{[1-\xi(t_{k})]}(\mathbf{Z}(t_{k});0,\mathbf{R}_{k}) \tag{8}\]  
**Assumption 2**. : _Based on Bayesian theory, the state vector is considered to obey the Gaussian distribution:_  
\[q[\mathbf{X}(t_{k})]=N\big{(}\mathbf{X}(t_{k});\mathbf{X}\big{(}t_{k}^{+}\big{)},\mathbf{P}\big{(}t_{k}^{+}\big{)}\big{)} \tag{9}\]  
_where \(N\big{(}\bullet;\mu,\delta^{2}\big{)}\) represents that the variable obeys the Gaussian distribution with mean \(\mu\) and covariance matrix \(\delta^{2}\)._  
**Assumption 3**. : _The inverse Wishart distribution is introduced to characterize the distributions of \(\mathbf{P}\big{(}t_{k}^{+}\big{)}\) and \(\mathbf{R}_{k}\); set \(q\big{[}\mathbf{P}\big{(}t_{k}^{+}\big{)}\big{]}=\mathrm{IW}\big{(}\mathbf{P} \big{(}t_{k}^{+}\big{)};t\big{(}t_{k}^{+}\big{)},\mathbf{T}\big{(}t_{k}^{+} \big{)}\big{)}\) and \(q(\mathbf{R}_{k})=\mathrm{IW}\big{(}\mathbf{R}_{k};u\big{(}t_{k}^{+}\big{)}, \mathbf{U}\big{(}t_{k}^{+}\big{)}\big{)}\); \(\mathrm{IW}(\bullet;\mu,\mathbf{V})\) means that the variable obeys the inverse Wishart distribution with degrees of freedom (DOF) \(n\) and inverse scale matrix \(\mathbf{V}\)._  
**Assumption 4**. : _Beta distribution is selected to approximate \(\mathrm{Pro}[\tau(t_{k})|\mathbf{Z}_{1k-1}]\) and \(\mathrm{Pro}[\tau(t_{k-1})|\mathbf{Z}_{1k-1}]\), so that the prior and posterior distributions of the acquisition probabilities \(\tau(t_{k})\) and \(\tau(t_{k-1})\) are guaranteed to follow the same form._  
\[\mathrm{Pro}[\tau(t_{k})|\mathbf{Z}_{1:k-1}]=\mathrm{B}\big{(}\tau(t_{k}^{+}); \alpha\big{(}t_{k}^{+}\big{)},\beta\big{(}t_{k}^{+}\big{)}\big{)} \tag{10}\]  
\[\mathrm{Pro}[\tau(t_{k-1})|\mathbf{Z}_{1:k-1}]=\mathrm{B}\big{(}\tau(t_{k-1}); \alpha(t_{k}),\beta(t_{k})\big{)} \tag{11}\]  
_where \(\mathrm{Pro}[\tau(t_{k})|\mathbf{Z}_{1:k-1}]=\int\{\mathrm{Pro}\,[\tau(t_{k})| \tau(t_{k-1})\,]\times\mathrm{Pro}[\tau(t_{k-1})|\mathbf{Z}_{1:k-1}]d\tau(t_{ k-1})\}\)._  
Because \(\mathrm{Pro}[\tau(t_{k})|\mathbf{Z}_{1:k-1}]\) cannot be obtained, the smoothing coefficient \(\rho\) can be used to characterize \(\mathrm{Pro}[\tau(t_{k-1})|\mathbf{Z}_{1:k-1}]\). \[\left\{ \begin{array}{l}\alpha\big{(}t_{k}^{+}\big{)}=\rho\alpha(t_{k})\\ \beta\big{(}t_{k}^{+}\big{)}=\rho\beta(t_{k})\end{array} ,\ \rho\in(0,1]\right. \tag{12}\]  
**Remark 1**. : _Set \(\Theta=\big{\{}\mathbf{X}(t_{k}),\mathbf{P}\big{(}t_{k}^{+}\big{)},\xi(t_{k}),\tau(t_{k})\big{\}}\), the joint PDF can be expanded as:_  
\[ \begin{array}{ll}\mathrm{Pro}[\Theta,\mathbf{Z}_{1:k}]&=\mathrm{Pro}[ \mathbf{Z}(t_{k})|\mathbf{X}(t_{k}),\xi(t_{k})]\,\mathrm{Pro}\big{[}\mathbf{X} (t_{k})\big{|}\mathbf{Z}_{1:k-1},\mathbf{P}\big{(}t_{k}^{+}\big{)}\big{]}\\ &\times\mathrm{Pro}\big{[}\mathbf{P}\big{(}t_{k}^{+}\big{)}|\mathbf{Z}_{1:k-1} \big{]}\mathrm{Pro}(\mathbf{R}_{k}|\mathbf{Z}_{1:k-1}\big{]}\\ &\times\mathrm{Pro}[\xi(t_{k})|\tau(t_{k})|\mathrm{Pro}[\tau(t_{k})|\mathbf{Z}_ {1:k-1}]\mathrm{Pro}\tau(\mathbf{Z}_{1:k-1})\end{array}  \tag{13}\]  
_Moreover, the joint posteriori PDF is denoted as:_  
\[