The time-series smoother

A whole posterior path finding equilibrium at once.

The local-level model is a bead chain. It is a random-walk state observed in noise, $x_{t+1} = x_t + \mathcal{N}(0, q^{-1})$, $y_t = x_t + \mathcal{N}(0, r^{-1})$. Each green bead $x_t$ rides a frictionless vertical rod at its time slot. It hangs from its observation by a measurement spring of stiffness $r$, and is tied to its neighbours by process-noise springs of stiffness $q$. The rods stop the chain contracting sideways. The springs have zero natural length, so the diagonal neighbour spring's energy splits into a constant plus $\tfrac12 q (x_{t+1}-x_t)^2$. The energy is then the negative log posterior of the whole path,

$$ E \;=\; \tfrac12 \sum_t r\,(x_t - y_t)^2 \;+\; \tfrac12 \sum_t q\,(x_{t+1} - x_t)^2, $$

so the chain's resting shape is the Kalman smoother. It is equally Whittaker-Henderson graduation with a first-difference penalty. Whittaker published this method in 1923 to smooth mortality tables. Economists reached the same idea decades later. The Hodrick-Prescott filter is this construction with a second-difference penalty instead of a first-difference one. The dashed curve is the exact smoother from a tridiagonal solve. The beads just obey $F = ma$.

Stiffen the chain and it pulls taut toward a constant. All the evidence is pooled. Slacken it and each bead chases its own observation. Drag an observation and the disturbance propagates a few springs down the chain and dies out. The smoother's influence is local. That is why filtering and smoothing agree except near the ends. Add an observation and the whole posterior path re-equilibrates to accommodate it.

chain stiffness $q$ obs. stiffness $r$

beads energy energy at exact smoother sliders are log-scale

In the language of the introduction, each link of the chain is one series spring. It is the prediction step bleeding precision. Each measurement spring adds its mass in parallel. Run along the chain left to right doing those two moves and you have the Kalman filter. Let the whole chain relax at once and you have the smoother. Same springs, different questions. Pin part of the chain and you are conditioning. That is a Schur complement, as at schur.microprediction.org.

The usual explanation the textbook derivation, in full

Write the local-level model as a linear Gaussian state space, with process variance $q^{-1}$ and measurement variance $r^{-1}$:

$$ x_{t+1} = x_t + w_t,\ w_t \sim \mathcal{N}(0, q^{-1}), \qquad y_t = x_t + v_t,\ v_t \sim \mathcal{N}(0, r^{-1}). $$

Run the Kalman filter forward. At each step predict, then correct with the new observation, carrying the estimate $\hat x_{t\mid t}$ and its variance $P_{t\mid t}$:

$$ \hat x_{t\mid t-1} = \hat x_{t-1\mid t-1}, \qquad P_{t\mid t-1} = P_{t-1\mid t-1} + q^{-1}, $$

$$ K_t = \frac{P_{t\mid t-1}}{P_{t\mid t-1} + r^{-1}}, \qquad \hat x_{t\mid t} = \hat x_{t\mid t-1} + K_t\big(y_t - \hat x_{t\mid t-1}\big), \qquad P_{t\mid t} = (1 - K_t)\,P_{t\mid t-1}. $$

The filtered estimate at time $t$ uses only the past. The smoother wants the future too, so make a second pass backward from the end. This is the Rauch-Tung-Striebel recursion, which corrects each filtered estimate using the already-smoothed estimate to its right:

$$ C_t = \frac{P_{t\mid t}}{P_{t+1\mid t}}, \qquad \hat x_{t\mid N} = \hat x_{t\mid t} + C_t\big(\hat x_{t+1\mid N} - \hat x_{t+1\mid t}\big). $$

Two passes and a stack of gains and variances, and $\hat x_{t\mid N}$ is the smoothed path. Because everything is Gaussian, that path is also the mode of the joint posterior, so it minimises the energy $E$ above. Setting the gradient of $E$ to zero gives the same answer in one batch, as the normal equations

$$ (r\,I + q\,L)\,x = r\,y, $$

where $L$ is the graph Laplacian of the chain. The matrix is tridiagonal, so a single Thomas sweep solves it. The forward-backward recursion and this linear solve return the identical path, by two different routes through the same bookkeeping.

The physics proof the same path, read off the chain

Two rules from earlier pages already do the work. The centre-of-mass rule from the fusion page fixes where a bead balances its evidence, at a precision-weighted mean. The reduced mass rule fixes how much precision survives a series link. Each bead feels one measurement spring and its two neighbour springs. It rests where those pulls cancel:

$$ r\,(y_t - x_t) + q\,(x_{t-1} - x_t) + q\,(x_{t+1} - x_t) = 0. $$

That is one row of $(rI + qL)x = ry$, read straight off the chain. Every bead asserts its own balance at the same instant. The rails hold the time slots apart and the whole chain relaxes to a single joint rest state. No forward pass, no backward recursion, just equilibrium.