The Kalman Filter

Summary

The Kalman filter is the closed-form solution to the Bayesian filtering equations for a linear-Gaussian state-space model. Every distribution stays Gaussian, so the filter propagates only a mean ${\mathbf{m}}_k$ and covariance ${\mathbf{P}}_k$ through a two-step recursion: a prediction step that pushes the state forward through the dynamics, and an update step that corrects it with the new measurement via the Kalman gain ${\mathbf{K}}_k$ and the innovation ${\mathbf{v}}_k$. It runs at constant cost per time step and is the forward pass feeding both The RTS Smoother and the marginal likelihood.

Overview

Bayesian filtering computes the filtering distribution $p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k})$ — the posterior of the current state given all measurements up to now. The general recursion (Särkkä Thm. 4.1) alternates a Chapman–Kolmogorov prediction and a Bayes-rule update. For linear-Gaussian models these reduce to matrix algebra on $({\mathbf{m}}_k,{\mathbf{P}}_k)$.

Main Content

Theorem: Bayesian filtering equations (Särkkä Thm. 4.1)

Starting from the prior $p({\mathbf{x}}_0)$, the predicted and filtering distributions obey, for $k=1,2,\dots$:

• Prediction step (Chapman–Kolmogorov equation): $$p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k-1})=\int p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1})p({\mathbf{x}}_{k-1}\mid {\mathbf{y}}_{1:k-1})\mathrm{d}{\mathbf{x}}_{k-1}.\text{\hspace{1em}(4.11)}$$
• Update step (Bayes’ rule): $$p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k})=\frac{1}{Z_k}p({\mathbf{y}}_k\mid {\mathbf{x}}_k)p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k-1}),\text{\hspace{1em}(4.12)}$$ $$Z_k=\int p({\mathbf{y}}_k\mid {\mathbf{x}}_k)p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k-1})\mathrm{d}{\mathbf{x}}_k.\text{\hspace{1em}(4.13)}$$

The normalizer $Z_k=p({\mathbf{y}}_k\mid {\mathbf{y}}_{1:k-1})$ is the one-step predictive density of the data — the building block of the marginal likelihood.

Theorem: Kalman filter (Särkkä Thm. 4.2)

For the linear-Gaussian model ${\mathbf{x}}_k={\mathbf{A}}_{k-1}{\mathbf{x}}_{k-1}+{\mathbf{q}}_{k-1}$, ${\mathbf{y}}_k={\mathbf{H}}_k{\mathbf{x}}_k+{\mathbf{r}}_k$, the filtering distributions are Gaussian,

$$p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k-1})=\mathcal{N}({\mathbf{x}}_k\mid {\mathbf{m}}_k^{-},{\mathbf{P}}_k^{-}),p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k})=\mathcal{N}({\mathbf{x}}_k\mid {\mathbf{m}}_k,{\mathbf{P}}_k),p({\mathbf{y}}_k\mid {\mathbf{y}}_{1:k-1})=\mathcal{N}({\mathbf{y}}_k\mid {\mathbf{H}}_k{\mathbf{m}}_k^{-},{\mathbf{S}}_k),\text{\hspace{1em}(4.19)}$$

computed by the following recursion, started from ${\mathbf{m}}_0,{\mathbf{P}}_0$.

Prediction step:

$${\mathbf{m}}_k^{-}={\mathbf{A}}_{k-1}{\mathbf{m}}_{k-1}$$ $${\mathbf{P}}_k^{-}={\mathbf{A}}_{k-1}{\mathbf{P}}_{k-1}{\mathbf{A}}_{k-1}^{\mathsf{T}}+{\mathbf{Q}}_{k-1}\text{\hspace{1em}(4.20)}$$

Update step:

$${\mathbf{v}}_k={\mathbf{y}}_k-{\mathbf{H}}_k{\mathbf{m}}_k^{-}\text{(innovation / measurement residual)}$$ $${\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_k^{-}{\mathbf{H}}_k^{\mathsf{T}}+{\mathbf{R}}_k\text{(innovation covariance)}$$ $${\mathbf{K}}_k={\mathbf{P}}_k^{-}{\mathbf{H}}_k^{\mathsf{T}}{\mathbf{S}}_k^{-1}\text{(Kalman gain)}$$ $${\mathbf{m}}_k={\mathbf{m}}_k^{-}+{\mathbf{K}}_k{\mathbf{v}}_k$$ $${\mathbf{P}}_k={\mathbf{P}}_k^{-}-{\mathbf{K}}_k{\mathbf{S}}_k{\mathbf{K}}_k^{\mathsf{T}}\text{\hspace{1em}(4.21)}$$

Reading the equations

• ${\mathbf{m}}_k^{-},{\mathbf{P}}_k^{-}$ are the predicted (“prior”) mean and covariance before seeing ${\mathbf{y}}_k$; the superscript $-$ marks “one step ahead, no current measurement.” ${\mathbf{m}}_k,{\mathbf{P}}_k$ are the updated (“posterior”) quantities after ${\mathbf{y}}_k$.
• The innovation ${\mathbf{v}}_k={\mathbf{y}}_k-{\mathbf{H}}_k{\mathbf{m}}_k^{-}$ is what the measurement tells you beyond the prediction; ${\mathbf{S}}_k$ is its covariance.
• The Kalman gain ${\mathbf{K}}_k$ is the optimal weight on the innovation: large when the prediction is uncertain (${\mathbf{P}}_k^{-}$ large) or the measurement is precise (${\mathbf{R}}_k$ small), small otherwise.
• The update always reduces covariance: ${\mathbf{P}}_k={\mathbf{P}}_k^{-}-{\mathbf{K}}_k{\mathbf{S}}_k{\mathbf{K}}_k^{\mathsf{T}}⪯{\mathbf{P}}_k^{-}$. An equivalent “information” form is ${\mathbf{P}}_k=(({\mathbf{P}}_k^{-}{)}^{-1}+{\mathbf{H}}_k^{\mathsf{T}}{\mathbf{R}}_k^{-1}{\mathbf{H}}_k{)}^{-1}$.
• Derivation (Särkkä §4.3): apply the Gaussian joint/conditioning lemmas (Lemmas A.1–A.2) to $p({\mathbf{x}}_{k-1},{\mathbf{x}}_k\mid {\mathbf{y}}_{1:k-1})$ for the prediction and to $p({\mathbf{x}}_k,{\mathbf{y}}_k\mid {\mathbf{y}}_{1:k-1})$ for the update.

Algorithm (per time step)

given m_{k-1}, P_{k-1}:
  # predict
  m⁻ = A m_{k-1}
  P⁻ = A P_{k-1} Aᵀ + Q
  # update with y_k
  v  = y_k − H m⁻
  S  = H P⁻ Hᵀ + R
  K  = P⁻ Hᵀ S⁻¹
  m_k = m⁻ + K v
  P_k = P⁻ − K S Kᵀ

Cost is constant per step (a few $n\times n$ / $m\times m$ products and one $m\times m$ inverse) — the key practical advantage of the recursive form over batch Bayes.

Examples

Kalman filter for the Gaussian random walk (Särkkä Ex. 4.2)

For the local-level model ($A=1$, $H=1$, scalar $Q,R$) the recursion collapses to scalars:

$$m_k^{-}=m_{k-1},P_k^{-}=P_{k-1}+Q,$$ $$m_k=m_k^{-}+\frac{P_k^{-}}{P_k^{-}+R}(y_k-m_k^{-}),P_k=P_k^{-}-\frac{(P_k^{-}{)}^2}{P_k^{-}+R}.\text{\hspace{1em}(4.31)}$$

The scalar gain $K_k=P_k^{-}/(P_k^{-}+R)$ interpolates between trusting the prediction ($R$ large $\Rightarrow K\to 0$) and trusting the measurement ($R$ small $\Rightarrow K\to 1$). This same filter is smoothed in The RTS Smoother.

Kalman filter for car tracking (Särkkä Ex. 4.3)

Running the 4-D constant-velocity model (see Linear-Gaussian State-Space Models) on noisy position measurements recovers the full position+velocity trajectory. In Särkkä’s simulation the position RMSE drops from $0.77$ (raw measurements) to $0.43$ (filter estimate) — the filter exploits the dynamics to denoise. The RTS smoother lowers it further to $0.27$.

Connections

• Linear-Gaussian State-Space Models — the model being filtered; defines $\mathbf{A},\mathbf{H},\mathbf{Q},\mathbf{R}$
• The RTS Smoother — backward pass that reuses ${\mathbf{m}}_k,{\mathbf{P}}_k,{\mathbf{m}}_{k+1}^{-},{\mathbf{P}}_{k+1}^{-}$ from this filter
• Marginal Likelihood via the Kalman Filter — the innovations ${\mathbf{v}}_k$ and covariances ${\mathbf{S}}_k$ produced here factor the likelihood
• State-Space Models and the Kalman Filter - Overview — pipeline context

See Also

• Bayesian Structural Time-Series Model — BSTS inference runs this filter (and its simulation-smoother variant) internally
• MCMC Inference for CausalImpact — Durbin–Koopman simulation smoother builds on Kalman filtering
• Local Linear Trend and Seasonality — the state components the filter tracks in practice