Kalman Filter Essence

There’s a myriad of ressources for the Kalman filter, so this is definitely not the place to write yet another post about it.

I just want to group a few facts together.

• The Kalman Filter is a Bayes Filter, and thus the following properties are “inherited”.
• The prediction step marginalizes out the prior state \(\mathbf{x}\), i.e., \(p({\textbf {x}}_{t}|{\textbf {z}}_{1:t-1})=\int p({\textbf {x}}_{t}|{\textbf {x}}_{t-1})p({\textbf {x}}_{t-1}|{\textbf {z}}_{1:t-1})\,d{\textbf {x}}_{t-1}\)
• The update state is a multiplication of the state distribution with the conditional observation distribution (ignoring the normalization), i.e., \(p({\textbf {x}}_{t}|{\textbf {z}}_{1:t})={\frac {p({\textbf {z}}_{t}|{\textbf {x}}_{t})p({\textbf {x}}_{t}|{\textbf {z}}_{1:t-1})}{p({\textbf {z}}_{t}|{\textbf {z}}_{1:t-1})}}\propto p({\textbf {z}}_{t}|{\textbf {x}}_{t})p({\textbf {x}}_{t}|{\textbf {z}}_{1:t-1})\)
• \(\mathbf{x} = \hat{\mathbf{x}} + \mathbf{K}\mathbf{\tilde{y}}\)
• \(\mathbf{x} = \hat{\mathbf{x}} + \hat{\mathbf{P}}\mathbf{H}^T(\mathbf{S}^{-1})\mathbf{\tilde{y}}\)
• \(\mathbf{x} = \hat{\mathbf{x}} + \hat{\mathbf{P}}\mathbf{H}^T(\mathbf{H}\hat{\mathbf{P}}\mathbf{H}^T + \mathbf{R})^{-1}\mathbf{\tilde{y}}\)
• Should be
• \(\mathbf{x} = (\mathbf{\Lambda}_x+\mathbf{\Lambda}_y)^{-1} (\mathbf{\Lambda}_x \mathbf{x} + \mathbf{\Lambda}_y \mathbf{\tilde{y}})\)