Reduced mass
Springs in series, or how noise drains the weight of evidence.
Two systems, same prior (dark body). On the top rail the observation reaches the test particle through a chain: a spring of stiffness $\phi$ to a free intermediate joint (the latent $x_2$), then a spring of stiffness $\rho$ onward — the hierarchical model $y \mid x_2 \sim \mathcal N(x_2, \phi^{-1})$, $x_2 \mid \mu \sim \mathcal N(\mu, \rho^{-1})$. On the bottom rail the chain has been consolidated into a single body wired directly to the particle, carrying the reduced mass $\tilde\phi = \phi\rho/(\phi+\rho)$ — the series-spring rule, and the mechanical face of “variances add along a chain”: $\phi^{-1} + \rho^{-1}$.
Both particles settle on the same dashed line: the test particle cannot tell the chain from its consolidation. Slacken $\rho$ and watch the bottom body shrink as the observation's effective weight drains away; at the far right the channel is rigid ($\rho = \infty$), the joint is pinned, and the full mass $\phi$ comes through. Drag any body, drag the particles off equilibrium and let go, or pluck both at once.
The physicists' reduced-mass formula $m_1 m_2/(m_1+m_2)$ and the statisticians' “precisions combine harmonically through a channel” are the same equation. Chain more joints and the rule telescopes: $\tilde\phi^{-1} = \sum_j \rho_j^{-1}$. The Kalman filter's prediction step is exactly one such link — process noise is a sloppy spring between yesterday's posterior and today's prior. Derivation in the introduction, steps 3–5.