← Demos

The filter through time

One body, waxing and waning: the Kalman filter as it actually runs, tick by tick.

The smoother relaxes an entire path at once. The filter is more like a single body of evolving mass, alternating the two moves from the introduction as time advances. Each tick:

1. Predict. The belief chains through a process-noise spring of stiffness $q = 1/Q$ — a series composition, so the mass shrinks: $\pi_{\text{pred}} = \pi q/(\pi+q)$, mean unchanged. Watch the body deflate.
2. Update. If an observation of mass $r=1/R$ arrives, it fuses in parallel — masses add, $\pi' = \pi_{\text{pred}} + r$ — and the body relaxes to the new centre of mass. Watch it re-inflate and slide.

Skip an observation and the body just keeps shrinking, unchecked — staleness with nothing to arrest it. Play to watch the filter run, or Step one tick at a time. The dashed grey line is the true hidden path; dots are observations (some withheld, per the slider); the solid line is the trail of filtered means.

Filtering is exactly this alternation, run left to right once. Smoothing (the landing-page demo) is what happens if, having reached the end, you let the whole chain relax again with every link present at once — future evidence now pulls on the past too. Same two moves, same springs; the only difference is whether the chain gets to see its own future before settling.