Reduced mass

Springs in series, or how noise drains the weight of evidence.

Two systems share the same prior, the dark body. On the top rail the observation reaches the test particle through a chain. A spring of stiffness $\phi$ runs to a free intermediate joint, the latent $x_2$. A second spring of stiffness $\rho$ carries on to the particle. That is 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. It carries the reduced mass $\tilde\phi = \phi\rho/(\phi+\rho)$. That is 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 at $\rho = \infty$. 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.

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

reduced mass $\tilde\phi$ chain particle consolidated particle closed form

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 to $\tilde\phi^{-1} = \sum_j \rho_j^{-1}$. The formula turns two orbiting bodies into one fictitious body about a fixed point. Newton solved that two-body problem by geometry in the Principia of 1687, long before anyone weighed evidence with it. The Kalman filter's prediction step is one such link. Process noise is a slack spring between yesterday's posterior and today's prior. Derivation in the introduction, steps 3–5.

The usual explanation marginalising the latent variable

The chain is a two-stage model. The observation $y$ sees the latent joint $x_2$, and the joint sees the target $\mu$:

$$ y \mid x_2 \sim \mathcal{N}(x_2, \phi^{-1}), \qquad x_2 \mid \mu \sim \mathcal{N}(\mu, \rho^{-1}). $$

To learn what $y$ says about $\mu$, integrate the nuisance $x_2$ out. Multiply the two densities and keep the exponent:

$$ p(y \mid \mu) \;\propto\; \int \exp\!\Big[ -\tfrac\phi2 (y-x_2)^2 - \tfrac\rho2 (x_2-\mu)^2 \Big]\, dx_2. $$

The exponent is quadratic in $x_2$, with leading coefficient $\phi+\rho$. Collect its powers and complete the square in $x_2$:

$$ -\tfrac12(\phi+\rho)\Big(x_2 - \tfrac{\phi y + \rho\mu}{\phi+\rho}\Big)^2 \;-\; \tfrac12\,\frac{\phi\rho}{\phi+\rho}(y-\mu)^2. $$

The first term integrates to a constant, since a Gaussian integral over $x_2$ does not depend on $y$. The second term is what survives. It is a Gaussian in $y$ whose variance is the sum of the two stage variances:

$$ y \mid \mu \sim \mathcal{N}\big(\mu,\; \phi^{-1} + \rho^{-1}\big), \qquad \tilde\phi = \big(\phi^{-1}+\rho^{-1}\big)^{-1} = \frac{\phi\rho}{\phi+\rho}. $$

So the effective precision with which $y$ pins $\mu$ is the reduced mass. Marginalising the latent variable is one Gaussian integral, and it lands on the harmonic combination.

The physics proof springs in series, compliances add

Look at the top rail. One force runs the length of the chain, from $y$ through the joint to the particle. The same force stretches both springs, so their extensions add. Extension per unit force is compliance, one over stiffness. Compliances in series add:

$$ \frac{1}{\tilde\phi} = \frac1\phi + \frac1\rho \qquad\Longrightarrow\qquad \tilde\phi = \frac{\phi\rho}{\phi+\rho}. $$

Compliance is variance, so variances add along the chain. That is the whole content of the integral above, read off the geometry of two springs end to end. The consolidated body on the bottom rail carries this $\tilde\phi$ and then fuses with the prior in parallel, which is the centre-of-mass rule. There is no nuisance variable to chase.