## Appendix A
It follows that  
\[\big{(}\mathsf{F}_{t-1|t-1}+\mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Omega} _{t}\mathsf{A}_{t}\big{)}^{-1} =\big{(}\mathsf{F}_{t-1|t-1}+\mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Theta }_{t}\mathsf{A}_{t}+\mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t}\mathsf{S}\mathbf{ \[PERSON]}_{t}\mathsf{S}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t}\mathsf{A}_{t}\big{)}^{-1}\] (15) \[=\big{(}\mathsf{B}_{t}+\mathsf{A}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t} \mathsf{S}_{t}\mathbf{\[PERSON]}_{t}\mathsf{S}_{t}^{\mathsf{T}}\mathbf{\Theta}_{t}\mathsf{A }_{t}\big{)}^{-1}\] (16) \[=\mathsf{B}_{t}^{-1}-\mathsf{B}_{t}^{-1}\mathsf{A}_{t}^{\mathsf{T} }\mathbf{\Theta}_{t}\mathsf{S}_{t}\big{(}\mathbf{\Psi}_{t}^{-1}+\mathsf{S}_{t}^{ \mathsf{T}}\mathbf{\Theta}_{t}\mathsf{A}_{t}\mathsf{B}_{t}^{-1}\mathsf{A}_{t}^{ \mathsf{T}}\mathbf{\Theta}_{t}\mathsf{S}_{t}\big{)}^{-1}\times\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\  
Next, consider the coefficient matrix of \(\mathbf{y}_{t}\) in Equation (A3):  
\[\big{(}C_{t}^{\mathsf{T}}\overline{\mathbf{v}}_{t}-\mathbf{\Sigma}_{ t}\big{)}+\big{(}\mathbf{\Omega}_{t}\mathbf{A}_{t}-C_{t}^{\mathsf{T}}\mathbf{\Sigma}_{t}^{ \mathsf{T}}\mathbf{A}_{t}\big{)}\big{(}F_{t-1|t-1}+A_{t}^{\mathsf{T}}\mathbf{ \Omega}_{t}\mathbf{A}_{t}\big{)}^{-1}\mathbf{A}_{t}^{\mathsf{T}}\mathbf{\Sigma }_{t}\] \[=\big{  
* (10) [PERSON]; [PERSON] Efficient simulation and integrated likelihood estimation in state space models. _Int. J. Math. Model. Numer. Optim._**2009**, \(1\), 101-120. [CrossRef]
* (11) [PERSON]; [PERSON] On Kalman filtering, posterior mode estimation and Fisher scoring in dynamic exponential family regression. _Metrika_**1991**, _38_, 37-60. [CrossRef]
* (12) [PERSON] Robust Methods for Kalman Filtering/Smoothing and Bundle Adjustment. Ph.D. Thesis, Department of Mathematics, University of Washington, Seattle, WA, USA, 2010. * (13) [PERSON]; [PERSON]; [PERSON] Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation. _arXiv_, **2013**, arXiv:1303.1993. * (14) [PERSON] The marginal likelihood for parameters in a discrete Gauss-Markov process. _IEEE Trans. Signal Process_**2000**, _48_, 870-873. [CrossRef] [PubMed]
* (15) [PERSON]; [PERSON] Least squares estimation of discrete linear dynamic systems using orthogonal transformations. _Siam J. Numer. Anal._**1977**, _14_, 180-193. [CrossRef]
* (16) [PERSON] The numerical solution of the Kalman filtering problem. In _Computational and Combinatorial Methods in System Theory_; [PERSON], [PERSON], Eds. ; Elsevier, Amsterdam, The Netherlands, 1986; pp. 23-36. * (17) [PERSON]; [PERSON] (Eds.) _Readings in Unobserved Component Models_; Oxford University Press: Oxford, UK, 2005. * (18) [PERSON]; [PERSON]; [PERSON]; [PERSON] _Time Series Analysis, Forecasting and Control_, 5 th ed. ; Wiley: Hoboken, NJ, USA, 2015. * (19) [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON] Optimal linear filter design for systems with correlation in the measurement matrices and noises: Recursive algorithm and applications. _Int. J. Syst. Sci._**2014**, _45_, 1548-1562. [CrossRef]
* (20) [PERSON]; [PERSON] _Kalman Filtering with Real-time Applications_, 5 th ed. ; Springer: Berlin/Heidelberg, Germany, 2017. * (21) [PERSON]; [PERSON] _Kalman Filtering: Theory and Practice Using MATLAB_, 4 th ed. ; Wiley: Hoboken, NJ, USA, 2015. * (22) [PERSON]; [PERSON]. _Introduction to Random Signals and Applied Kalman Filtering_, 4 th ed. ; Wiley: Hoboken, NJ, USA, 2012. * (23) [PERSON] _Advanced Kalman Filtering, Least-Squares and Modeling_; Wiley: Hoboken, NJ, USA, 2011. * (24) [PERSON]; [PERSON] _Optimal Filtering_; Prentice-Hall: Englewood Cliffs, NJ, USA, 1979. * (25) [PERSON] _Multivariate Time Series with Linear State Space Structure_; Springer: Berlin/Heidelberg, Germany, 2016. * (26) [PERSON] Topics in advanced time series analysis. In _Lectures in Probability and Statistics_; [PERSON], [PERSON], Eds.