Evidence fusion

The posterior mean is where the pulling stops.

The dark body is the prior; green bodies are observations. Each is a mass equal to its precision, tied by a spring to the white test particle, which relaxes to the centre of mass — the posterior mean. The bowl above is the potential energy, i.e. the negative log posterior; the particle is a ball rolling to its bottom. Drag any body, or add an observation and watch equilibrium re-form. The $\rho$ dial threads every observation through a noisy channel of that precision, replacing its mass $\phi$ by the reduced mass $\phi\rho/(\phi+\rho)$; at the far right the channel is rigid ($\rho=\infty$) and the update is the vanilla Kalman step.

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

posterior mean posterior precision (total mass) particle at

Everything on screen is computed twice: the readout uses the closed-form update $m' = \big(P^{-1}m + \sum_i \tilde\phi_i y_i\big)\big/\big(P^{-1} + \sum_i \tilde\phi_i\big)$, while the particle just integrates $F = ma$ with damping. That they agree is the point. Derivation in the introduction.