## 2 State Space Models  
Let \(\mathbf{y}_{t}\) denote a given time series of data vectors, where the subscript \(t=1,\ldots,N\) denotes a discrete time; let \(n\) denote the dimensions of the data vectors. The given time series shall be modelled by a linear state space model given by a pair of equations, the first of which is known as the observation equation, given by  
\[\mathbf{y}_{t}=\mathbf{C}_{t}\mathbf{x}_{t}+\boldsymbol{\epsilon}_{t} \tag{1}\]  
where \(\mathbf{x}_{t}\) denotes a series of state vectors, with \(m\) denoting the dimensions of the state vectors, \(\mathbf{C}_{t}\) denotes an \((n\times m)\)-dimensional matrix, known as the observation matrix, and \(\boldsymbol{\epsilon}_{t}\) denotes a series of noise vectors, known as observation noise, which are assumed to be drawn from a Gaussian distribution with a mean vector of zero and an \((n\times n)\)-dimensional covariance matrix, \(\mathsf{R}_{t}\). As the subscript \(t\) indicates, the observation matrix and the covariance matrix of the observation noise may depend on time. The second equation is known as the dynamical equation; it can be formulated in two different variants [17], the first, known as the \"contemporaneous form\", being given by  
\[\mathbf{x}_{t}=\mathsf{A}_{t}\mathbf{x}_{t-1}+\boldsymbol{\eta}_{t} \tag{2}\]  
while the second variant, known as the \"future form\", is given by  
\[\mathbf{x}_{t}=\mathsf{A}_{t}\mathbf{x}_{t-1}+\boldsymbol{\eta}_{t-1} \tag{3}\]  
where \(\mathsf{A}_{t}\) denotes an \((m\times m)\)-dimensional matrix, known as the state transition matrix, and \(\boldsymbol{\eta}_{t}\) denotes a series of noise vectors, known as dynamical noise, which are assumed to be drawn from a Gaussian distribution with a mean vector of zero and an \((m\times m)\)-dimensional covariance matrix, \(\mathsf{Q}_{t}\). As the subscript \(t\) indicates, the state transition matrix and the covariance matrix of the dynamical noise may also depend on time. In the future-form dynamical equation, Equation (3), many authors choose to write the time dependence of the state transition matrix as \(\mathsf{A}_{t-1}\) instead of \(\mathsf{A}_{t}\), but this is merely a matter of definition. The two noise terms, \(\boldsymbol{\epsilon}_{t}\) and \(\boldsymbol{\eta}_{t}\), may be correlated, i.e., the expectation of the product, \(\boldsymbol{\eta}_{t}\boldsymbol{\epsilon}_{t}^{T}\), may not vanish, such that an \((m\times n)\)-dimensional cross-covariance matrix needs to be defined by  
\[\mathsf{S}_{t}=\mathbb{E}(\boldsymbol{\eta}_{\tau}\boldsymbol{\epsilon}_{ \tau}^{T}) \tag{4}\]  
where the symbol \(\mathbb{E}\) denotes the expectation (over time \(\tau\)); this cross-covariance matrix may also depend on time. Then, the covariance matrix of the stacked vector of both noise terms is given by  
\[\mathbb{E}\bigg{(} \begin{bmatrix}\boldsymbol{\eta}_{\tau}\\ \boldsymbol{\epsilon}_{\tau}\end{bmatrix}\begin{bmatrix}\boldsymbol{\eta}_{ \tau}\\ \boldsymbol{\epsilon}_{\tau}\end{bmatrix} ^{T}\bigg{)}= \begin{pmatrix}\mathsf{Q}_ {t}&\mathsf{S}_{t}\\ \mathsf{S}_{t}^{T}&\mathsf{R}_{t}\end{pmatrix}  \tag{5}\]  
For later use, this cross-covariance matrix will now be inverted. First, new symbols for the inverses of the covariance matrices \(\mathsf{R}_{t}\) and \(\mathsf{Q}_{t}\) are introduced:  
\[\boldsymbol{\Phi}_{t}:=\mathsf{R}_{t}^{-1}\,,\quad\boldsymbol{\Theta}_{t}:= \mathsf{Q}_{t}^{-1} \tag{6}\]Note that \(\mathbf{\Phi}_{t}\) and \(\mathbf{\Theta}_{t}\) represent information matrices. The desired inverse matrix is formulated as  
\[ \begin{pmatrix}\mathsf{Q}_{t}&\mathsf{S}_{t}\\ \mathsf{S}_{t}^{\mathsf{T}}&\mathsf{R}_{t}\end{pmatrix} ^{-1}= \begin{pmatrix} \Omega_{t}&\mathbf{\Sigma}_{t}\\ \mathbf{\Sigma}_{t}^{\mathsf{T}}&\mathbf{\Psi}_{t}\end{pmatrix}