## 5 Weighted Least-Squares Approach to State Estimation
### Interpretation of the Filters and Smoothers  
The next required task is interpreting the filters and smoothers that were derived in the previous section. By construction, the output of the backward smoothing recursions, simply denoted by \(\mathbf{\mathsf{x}}_{t}\) in Equations (73) and (82), represents the smoothed state estimates, \(\mathbf{\mathsf{x}}_{t|N}\), and from Equation (67), it is also clear that the recursion of Equations (74) and (83) describes the covariance matrix of the smoothed state estimates. However, the meaning of the intermediate quantities \(\mathbf{\mathsf{Z}}_{t}\) and \(\mathbf{\mathsf{z}}_{t}\), to be computed by Equations (71), (72), (89), and (81), is not immediately obvious. In the case of non-correlated noise terms, \(\mathbf{\mathsf{z}}_{t}\) would directly correspond to the filtered information state, \(\mathbf{\mathsf{x}}_{t|t}\), but in the general case, this is no longer true. The condition for interpreting the quantities \(\mathsf{Z}_{t}\) and \(\mathbf{z}_{t}\) is that the known information filters, as reviewed above in Section 3, be reproduced. This condition leads to the following interpretations: for the contemporaneous form,  
\[\mathsf{Z}_{t} = \begin{cases}\mathsf{F}_{t|t}+\mathsf{A}_{t+1}^{\mathsf{T}} \boldsymbol{\Omega}_{t+1}\mathbf{A}_{t+1}&\text{if }\,t=1,\ldots,N-1\\ \mathsf{F}_{t|t}&\text{if }\,t=N\end{cases}  \tag{84}\] \[\mathbf{z}_{t} = \begin{cases}\boldsymbol{\chi}_{t|t}+\mathsf{A}_{t+1}^{\mathsf{ T}}\boldsymbol{\Sigma}_{t+1}\mathbf{y}_{t+1}&\text{if }\,t=1,\ldots,N-1\\ \boldsymbol{\chi}_{t|t}&\text{if }\,t=N\end{cases}  \tag{85}\]  
and for the future form,  
\[\mathsf{Z}_{t} = \begin{cases}\mathsf{F}_{t|t}+\bar{\mathsf{A}}_{t+1}^{\mathsf{ T}}\boldsymbol{\Omega}_{t}\bar{\Lambda}_{t+1}&\text{if }\,t=1,\ldots,N-1\\ \mathsf{F}_{t|t}&\text{if }\,t=N\end{cases}  \tag{86}\] \[\mathbf{z}_{t} = \begin{cases}\boldsymbol{\chi}_{t|t}+\bar{\mathsf{A}}_{t+1}^{ \mathsf{T}}\boldsymbol{\Sigma}_{t}\mathbf{y}_{t}&\text{if }\,t=1,\ldots,N-1\\ \boldsymbol{\chi}_{t|t}&\text{if }\,t=N\end{cases}  \tag{87}\]  
Equation (84) has already been given by [PERSON] and [PERSON] [11], although only for the non-correlated case, \(\mathsf{S}_{t}=0\). Then, in the case of the contemporaneous form, the information filter becomes, for \(t=2,\ldots,N\),  
\[\boldsymbol{\chi}_{t|t} =\bigl{(}\mathsf{C}_{t}^{\mathsf{T}}\boldsymbol{\Psi}_{t}- \boldsymbol{\Sigma}_{t}\bigr{)}\mathbf{y}_{t}\] \[\qquad\qquad-\bigl{(}\mathsf{A}_{t}  
and the smoother becomes, for \(t=N-1,\ldots,1\),  
\[\mathbf{x}_{t|N} =\big{(}\mathsf{F}_{t|t}+\bar{\mathsf{A}}_{t+1}^{\mathsf{T}}\mathbf{ \Omega}_{t}\bar{\mathsf{A}}_{t+1}\big{)}^{-1}\Big{(}\mathbf{\chi}_{t|t}+\bar{ \mathsf{A}}_{t+1}^{\mathsf{T}}\mathbf{\Sigma}_{t}\mathbf{y}_{t}-\big{(}\mathsf{C}_{t}^ {\mathsf{T}}\mathbf{\Sigma}_{t}^{\mathsf{T}}-\bar{\mathsf{A}}_{t+1}^{\mathsf{T}}\bm {\Omega}_{t}\big{)}\mathbf{x}_{t+1|N}\Big{)} \tag{94}\] \[\mathsf{P}_{t|N} =\big{(}\mathsf{F}_{t|t}+\bar{\mathsf{A}}_{t+1}^{\mathsf{T}}\mathbf{ \Omega}_{t}\bar{\mathsf{A}}_{t+1}\big{)}^{-1}+\big{(}\mathsf{F}_{t|t}+\bar{ \mathsf{A}}_{t+1}^{\mathsf{T}}\mathbf{\Omega}_{t}\bar{\mathsf{A}}_{t+1}\big{)}^{-1}\times\] \[\quad\big{(}\mathsf{C}_{t}^{\mathsf{T}}\mathbf{\Sigma}_{t}^{\mathsf{T }}+\mathsf{A}_{t+1}^{\mathsf{T}}\mathbf{\Omega}_{t}\big{)}\mathsf{P}_{t+1|N}\big{(} \mathsf{C}_{t}^{\mathsf{T}}\mathbf{\Sigma}_{t}^{\mathsf{T}}+\mathsf{A}_{t+1}^{ \mathsf{T}}\mathbf{\Omega}_{t}\big{)}^{\mathsf{T}}\big{(}\mathsf{F}_{t|t}+\bar{ \mathsf{A}}_{t+1}^{\mathsf{T}}\mathbf{\Omega}_{t}\bar{\mathsf{A}}_{t+1}\big{)}^{-1} \tag{95}\]  
The modifications for the limit case \(t=1\) follow accordingly from Equations (50), (51), (77), and (79); the case \(t=N\) requires no modifications to the forward filter pass.