Index: State-Space and Kalman Filter

State-Space Models and the Kalman Filter

Routing Summary

This folder fills the vault’s state-space / Kalman gap (vault gap #13), ingesting the linear-Gaussian core of Särkkä (2013), Bayesian Filtering and Smoothing. It explains the machinery that Bayesian Structural Time-Series Model and Local Linear Trend and Seasonality use but never derive. Contains 5 notes.

• New here / want the big picture? → State-Space Models and the Kalman Filter - Overview
• Need the model definition (observation + transition equations, symbols)? → Linear-Gaussian State-Space Models
• Need the predict/update recursion, Kalman gain, innovations? → The Kalman Filter
• Need the backward smoothing recursion / forward-backward pass? → The RTS Smoother
• Need the likelihood for MCMC / parameter estimation? → Marginal Likelihood via the Kalman Filter

Concept Map

Concept	Note	Type	Depends On	Key Result
Pipeline overview	State-Space Models and the Kalman Filter - Overview	overview	—	Predict→update→smooth→likelihood; filtering Eqs (4.11)-(4.13)
Linear-Gaussian model	Linear-Gaussian State-Space Models	definition	State-Space Models and the Kalman Filter - Overview	Eqs (4.17)-(4.18): ${\mathbf{x}}_k=\mathbf{A}{\mathbf{x}}_{k-1}+\mathbf{q}$, ${\mathbf{y}}_k=\mathbf{H}{\mathbf{x}}_k+\mathbf{r}$; Markov props (4.2),(4.4)
Kalman filter	The Kalman Filter	theorem	Linear-Gaussian State-Space Models	Eqs (4.20)-(4.21): predict ${\mathbf{m}}_k^{-},{\mathbf{P}}_k^{-}$; gain ${\mathbf{K}}_k$; update ${\mathbf{m}}_k,{\mathbf{P}}_k$
RTS smoother	The RTS Smoother	theorem	The Kalman Filter	Eq (8.6): backward gain ${\mathbf{G}}_k$, ${\mathbf{m}}_k^s,{\mathbf{P}}_k^s$; ${\mathbf{P}}_k^s⪯{\mathbf{P}}_k$
Marginal likelihood	Marginal Likelihood via the Kalman Filter	concept	The Kalman Filter	Eq (12.5) prediction-error decomp; energy fn (12.38); enables MCMC

Concept Dependency Chain

State-Space Models and the Kalman Filter - Overview
  └── Linear-Gaussian State-Space Models  (Eqs 4.17-4.18; Markov 4.2,4.4)
        └── The Kalman Filter  (forward: predict 4.20, update 4.21)
              ├── The RTS Smoother  (backward: G_k, m_k^s, P_k^s, Eq 8.6)
              └── Marginal Likelihood via the Kalman Filter
                    ├── Prediction-error decomposition (Eq 12.5)
                    ├── Energy function recursion (Eq 12.38) via v_k, S_k
                    └── MAP / MCMC / EM over θ → Bayesian Structural Time-Series Model

Notes

• State-Space Models and the Kalman Filter - Overview — CONTAINS: the predict→update→smooth→likelihood pipeline; general filtering Eqs (4.11)-(4.13); why state-space form gives constant cost, modularity, tractable likelihood; running random-walk example
• Linear-Gaussian State-Space Models — CONTAINS: probabilistic state-space model (Def 4.1); Markov + conditional-independence properties (4.2, 4.4); linear-Gaussian model (4.17-4.18); full symbol glossary; random-walk and car-tracking examples with $\mathbf{A},\mathbf{H},\mathbf{Q},\mathbf{R}$ matrices
• The Kalman Filter — CONTAINS: Bayesian filtering equations (Thm 4.1); Kalman filter (Thm 4.2) full predict (4.20) + update (4.21) equations; innovation ${\mathbf{v}}_k$, innovation covariance ${\mathbf{S}}_k$, Kalman gain ${\mathbf{K}}_k$; algorithm pseudocode; scalar random-walk (4.31) and car-tracking examples
• The RTS Smoother — CONTAINS: Bayesian smoothing equations (Thm 8.1); RTS smoother (Thm 8.2) full backward recursion (8.6) with smoother gain ${\mathbf{G}}_k$; ${\mathbf{P}}_k^s⪯{\mathbf{P}}_k$; two-filter alternative; algorithm pseudocode; random-walk (8.16) and car-tracking examples
• Marginal Likelihood via the Kalman Filter — CONTAINS: prediction-error decomposition (Thm 12.1, Eq 12.5); energy function recursion for linear-Gaussian model (Thm 12.3, Eq 12.38); one-step predictive Gaussian (12.41); MAP/ML/MCMC/EM/Laplace; Metropolis-Hastings acceptance (12.17); noise-variance posterior example

Sources

• Sarkka 2013 - Bayesian Filtering and Smoothing.pdf — Särkkä S. 2013. Bayesian Filtering and Smoothing. Cambridge University Press. Chapters 3-4 (filtering & Kalman filter), 8 (smoothing & RTS), 12 (parameter estimation).

Cross-Links to Existing Vault Notes

• Bayesian Structural Time-Series Model — the applied state-space model whose observation/state equations (2.1-2.2) are a direct instance of (4.17); inference uses the Kalman/smoothing machinery here
• Local Linear Trend and Seasonality — concrete state components (level, slope, seasonal) assembled as block-diagonal $\mathbf{A},\mathbf{Q}$ and concatenated $\mathbf{H}$
• MCMC Inference for CausalImpact — Durbin-Koopman simulation smoother is the stochastic counterpart of the RTS smoother
• Single Marketing Time Series — ARIMA models have equivalent state-space representations; likelihoods evaluated via the Kalman filter
• Carryover Effects and Distributed Lags — AR / distributed-lag dynamics expressible in state-space form
• Hilbert Space Gaussian Processes — temporal GPs admit linear-Gaussian state-space (Kalman) representations