Schur damping

How far do you trust a floating joint's recoil?

Eliminating the free joint from the reduced mass demo's chain is a genuine Schur complement. The joint energy's Hessian in (particle $z$, joint $w$) is the precision matrix, and minimising out $w$ takes the Schur complement of the $w$ block:

$$ \Lambda = \begin{pmatrix} p+\rho & -\rho \\ -\rho & \rho+\phi \end{pmatrix}, \qquad (p+\rho) \;-\; \frac{\rho^2}{\rho+\phi} \;=\; p \;+\; \underbrace{\frac{\phi\rho}{\phi+\rho}}_{S(1)} . $$

The prior's $p$ rides through untouched, so the particle's effective stiffness toward the observation is $S(1) = \rho - \rho^2/(\rho+\phi)$: the channel stiffness minus a correction fed back by the joint's recoil. Schur damping keeps only a fraction $\gamma$ of that correction,

$$ S(\gamma) \;=\; \rho \;-\; \gamma\,\frac{\rho^2}{\rho+\phi}, $$

and the dial has a mechanical reading. At $\gamma = 1$ the joint floats free and the particle feels the exact marginal — the reduced mass. At $\gamma = 0$ the joint is clamped to the data: the particle inherits the channel's full stiffness $\rho$ and is pulled hard toward $y$, overconfident because the joint's own uncertainty has been ignored. Interior $\gamma$ trusts the recoil partially — the right posture when the coupling $\rho$ is estimated rather than known. The same dial, damping the off-diagonal feedback of a partitioned precision matrix, interpolates hierarchical and optimisation-based portfolio construction at the sister site, schur.microprediction.org.

The top rail is the true chain; the bottom rail is the $\gamma$-damped consolidation. Slide $\gamma$ to 1 and the two particles meet on the dashed line; slide it toward 0 and the bottom body swells to mass $\rho$ and drags its particle toward the observation. Drag any body to move the anchors.

damping $\gamma$ obs. mass $\phi$ channel $\rho$ prior mass $P^{-1}$

damped mass $S(\gamma)$ exact (reduced) mass $S(1)$ chain particle damped particle

Everything is still exact — the damped rail is a real spring system, just one whose stiffness toward the data is $S(\gamma)$ rather than $S(1)$. One honest caveat: the algebra fixes the damped stiffness, but where the damped spring anchors is a modelling choice; here the undamped remainder stays anchored at the data, which is the plug-in (clamped-joint) reading of $\gamma = 0$. Damping does not break the physics; it changes which physics you assert. See the Schur damping papers via the schur site's bibliography.