State-Space Models and the Kalman Filter - Overview

Summary

A state-space model describes a time series through a hidden Markov state that evolves over time (the transition equation) and is observed only noisily (the observation equation). Bayesian filtering computes the posterior of the state given past data via a recursive predict → update cycle; for the linear-Gaussian case this cycle has a closed form — the Kalman filter. The backward RTS smoother refines those estimates using all the data, and the filter’s by-product, the prediction-error decomposition, yields the marginal likelihood that makes MCMC over model parameters tractable. This is the machinery underneath Bayesian Structural Time-Series Model.

Overview

This folder ingests the linear-Gaussian core of Särkkä (2013), Bayesian Filtering and Smoothing. The four companion notes build a single recursive pipeline:

1. Linear-Gaussian State-Space Models — the model: a Markovian latent state with a linear transition and a linear-Gaussian observation. Defines every symbol.
2. The Kalman Filter — the forward recursion: predict (Chapman–Kolmogorov) then update (Bayes’ rule), giving the filtering distribution $p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k})$ in closed form.
3. The RTS Smoother — the backward recursion: a second pass that conditions each state on the entire data record ${\mathbf{y}}_{1:T}$, giving $p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:T})$.
4. Marginal Likelihood via the Kalman Filter — the filter’s innovations ${\mathbf{v}}_k$ and their covariances ${\mathbf{S}}_k$ factor the likelihood $p({\mathbf{y}}_{1:T}\mid \mathbf{\theta })$, enabling MAP estimation, MCMC, and EM over parameters $\mathbf{\theta }$.

Main Content

The recursive pipeline

The general probabilistic state-space model (Särkkä Def. 4.1) is

$${\mathbf{x}}_k\sim p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1}),{\mathbf{y}}_k\sim p({\mathbf{y}}_k\mid {\mathbf{x}}_k).$$

The Bayesian filtering equations (Särkkä Thm. 4.1) compute the filtering distribution recursively:

• Predict (Chapman–Kolmogorov): $p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k-1})=\int p({\mathbf{x}}_k\mid {\mathbf{x}}_{k-1})p({\mathbf{x}}_{k-1}\mid {\mathbf{y}}_{1:k-1})\mathrm{d}{\mathbf{x}}_{k-1}$
• Update (Bayes): $p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k})=\frac{1}{Z_k}p({\mathbf{y}}_k\mid {\mathbf{x}}_k)p({\mathbf{x}}_k\mid {\mathbf{y}}_{1:k-1})$

When the model is linear-Gaussian, every distribution stays Gaussian and these integrals collapse to matrix algebra — the Kalman filter (forward) and RTS smoother (backward).

Why state-space form matters

• Constant cost per step. Naïve Bayes over the full joint $p({\mathbf{x}}_{0:T}\mid {\mathbf{y}}_{1:T})$ costs more at every new observation; the recursive filter does $O(1)$ work per time step.
• Modularity. Independent state components (level, slope, seasonal, regression) stack as block-diagonal $\mathbf{A}$, $\mathbf{Q}$ and concatenated $\mathbf{H}$ — exactly how Local Linear Trend and Seasonality assembles a BSTS model.
• Tractable likelihood. The prediction-error decomposition (see Marginal Likelihood via the Kalman Filter) gives $p({\mathbf{y}}_{1:T}\mid \mathbf{\theta })$ analytically, so parameter inference (MAP / MCMC / EM) needs only the filter.

Examples

The running example across these notes is the Gaussian random walk plus noise (local level model), Särkkä Examples 4.1, 4.2, 8.1, 12.1:

$$x_k=x_{k-1}+q_{k-1},q_{k-1}\sim \mathcal{N}(0,Q);y_k=x_k+r_k,r_k\sim \mathcal{N}(0,R).$$

It is filtered in The Kalman Filter, smoothed in The RTS Smoother, and has its noise variance $R$ inferred in Marginal Likelihood via the Kalman Filter. This is the simplest BSTS state component.

Connections

• Linear-Gaussian State-Space Models — formal model definition
• The Kalman Filter — forward predict/update recursion
• The RTS Smoother — backward smoothing recursion
• Marginal Likelihood via the Kalman Filter — likelihood and parameter inference
• Bayesian Structural Time-Series Model — the applied state-space model this machinery powers
• Local Linear Trend and Seasonality — concrete state components in Kalman form

See Also

• MCMC Inference for CausalImpact — uses a simulation smoother (Durbin–Koopman) closely related to the RTS pass
• Single Marketing Time Series — ARIMA-based univariate alternative
• Hilbert Space Gaussian Processes — GPs admit equivalent state-space (Kalman) representations