The Dictionary
Mechanics on the left, statistics on the right. Exact whenever energies are quadratic.
| Mechanics | Statistics | |
|---|---|---|
| mass / spring stiffness $k$ | precision $1/\sigma^2$ | |
| potential energy $\tfrac12 k (z-x)^2$ | negative log-likelihood | |
| force $k(x-z)$ | score $\tfrac{d}{dz}\log p(z)$ | |
| equilibrium | posterior mean (= mode) | |
| centre of mass | Kalman update $m' = \dfrac{P^{-1}m + R^{-1}y}{P^{-1}+R^{-1}}$ | demo |
| masses in parallel add | independent precisions add | demo |
| springs in series: reduced mass $\dfrac{m_1 m_2}{m_1+m_2}$ | variances add through a noisy channel; the Kalman prediction step | demo |
| rigid rod on vertical springs | least squares regression | demo |
| rod on frictionless sliding collars | PCA (total least squares) | demo |
| bead chain: measurement + neighbour springs | Kalman smoother; Whittaker graduation | demo |
| masses re-weighted by error correlations | GLS; bid-ask bounce as anti-redundant evidence | demo |
| bead between corner posts; repulsive spring | minimum-variance portfolio; short selling | demo |
| one body, mass shrinking then regrowing over time | the Kalman filter, tick by tick | demo |
| pinning a node in a spring network | conditioning; Schur complement of a graphical model | demo |
| constant-tension pulley vs. Hookean spring | Lasso vs. Ridge; soft-thresholding | demo |
| thermal jitter | posterior sampling (Langevin) |
The correspondence is exact only for Gaussians: non-Gaussian likelihoods are non-Hookean springs — equilibrium still finds the mode, but subsystems can no longer be replaced by a single equivalent body.