Filter, window, or current-state solve

For the local Gaussian approximation, $P\approx H_{\rm total}^{-1}$; take marginal covariance blocks after solving the full system. Priors are already included in $H_{\rm total}$. See the derivation and implementation limits.

Key equations

Warm-up, from memory

$$ \Sigma_{i,k+1} = \mathbf{A}_k\,\Sigma_{ik}\,\mathbf{A}_k^{\top} + \mathbf{B}_k\,\Sigma_{\eta}\,\mathbf{B}_k^{\top}, \qquad \Sigma_{ii} = \mathbf{0} $$

Marginalization, by hand

$$ \begin{bmatrix}\mathbf{H}_{mm} & \mathbf{H}_{mk}\\ \mathbf{H}_{km} & \mathbf{H}_{kk}\end{bmatrix} \begin{bmatrix}\mathbf{x}_m\\ \mathbf{x}_k\end{bmatrix} = \begin{bmatrix}\mathbf{b}_m\\ \mathbf{b}_k\end{bmatrix} $$

$$ \begin{aligned} \mathbf{H}' &= \mathbf{H}_{kk} - \mathbf{H}_{km}\,\mathbf{H}_{mm}^{-1}\,\mathbf{H}_{mk}\\ \mathbf{b}' &= \mathbf{b}_{k} - \mathbf{H}_{km}\,\mathbf{H}_{mm}^{-1}\,\mathbf{b}_{m} \end{aligned} $$

$$ \mathbf{H} = \begin{bmatrix}4 & 2\\ 2 & 3\end{bmatrix}, \qquad \mathbf{b} = \begin{bmatrix}2\\ 1\end{bmatrix} $$

Practice

Explain the idea

A filter summarizes past measurements in the current state distribution and cannot relinearize old states. A sliding window retains recent states and can refine them together; marginalizing its oldest state preserves a local linearized prior but may create dense couplings. A current-state solve can also account for an uncertain previous state within a factor. Dropping measurements loses information, while fixing an uncertain estimate as exact can make the result overconfident.