Bayesian Filtering and Smoothing

Authors: Simo Särkkä Year: 2013 Tags: bayesian-inference, kalman-filtering, particle-filtering, state-space-models, sequential-monte-carlo, parameter-estimation

TL;DR

Concise graduate textbook unifying nonlinear Kalman filtering (EKF, UKF, GHKF, CKF), particle filtering and smoothing, and parameter estimation under a single Bayesian probabilistic framework. Intended for advanced undergraduates, graduate students, and practitioners needing enough depth to implement the methods. Published as volume 3 in the Cambridge IMS Textbooks series.

First pass — the five C's

Category. Pedagogical textbook / unified survey of algorithms for sequential Bayesian inference.

Context. Statistical signal processing and state estimation subfield. Directly builds on: Kalman (1960) — original discrete-time linear filter; Rauch, Tung & Striebel (1965) — fixed-interval smoother; Jazwinski (1970) — Bayesian probabilistic treatment of nonlinear filtering; Stratonovich (1950s–60s) — earliest Bayesian formulation of nonlinear filtering, predating Kalman.

Correctness. Central load-bearing assumptions: (1) system admits a Markov state-space representation with known transition density p(x_k | x_{k−1}) and measurement density p(y_k | x_k); (2) Gaussian approximations are adequate for EKF/UKF-class methods; (3) Monte Carlo approximations converge as particle count grows. All three are explicitly stated by the book, though the second is an approximation the book acknowledges.

Contributions. - Derives all covered algorithms (KF, EKF, UKF, GHKF, CKF, RTS smoothers, particle filters/smoothers, EM/MCMC parameter estimation) from a single coherent Bayesian notation, making cross-method comparison explicit. - Introduces and unifies the general Gaussian filter/smoother framework from which GHKF, CKF, and sigma-point methods emerge as special cases. - Presents fixed-point and fixed-lag Gaussian smoother equations in the general Gaussian framework (Chapter 10). - Provides MATLAB exercises and companion code covering the full algorithm stack.

Clarity. Writing is deliberately terse and well-organized; the progressive generalization from batch linear regression to full nonlinear particle smoothing is a pedagogical strength, though the symbols table (xiii–xx) is unusually large and hints at notational overhead readers must manage.

Second pass — content

Main thrust: Sequential Bayesian inference over hidden Markov state-space models is intractable in full joint form but tractable as marginal filtering/smoothing distributions; the book systematically derives the spectrum of algorithms — from exact linear (KF/RTS) through Gaussian-approximation (EKF, UKF, cubature) to Monte Carlo (particle filters/smoothers) — and shows how each serves parameter estimation via EM, ML, MAP, or MCMC.

Supporting evidence: - Resonator example (Figures 1.6, 1.8, 1.9): discrete-time resonator with 2D state (position, velocity); filtering and smoothing means shown with 95% quantile bands — the only quantitative illustration in the provided text; no numerical RMSE values reported. - Theoretical argument: marginalizing over individual time-step distributions (Equations 1.4–1.6) reduces per-step computation to O(1) rather than growing with T, making long time series tractable. - Chapter 12 derives likelihood p(θ | y_{1:T}) ∝ p(θ) ∏k p(y_k | y, θ) as a by-product of running the filter at fixed parameters (Equation 1.8). - Not stated: comparative benchmark numbers (e.g., RMSE, runtime) between EKF, UKF, CKF, or particle filter on any standard dataset. - Not stated: convergence rates or sample-complexity results for particle methods.

Figures & tables: Chapter 1 contains Figures 1.1–1.9 (GPS schematic, radar tracking diagrams, brain imaging illustration, resonator signal/filter/smoother plots). Resonator figures (1.8, 1.9) have labeled axes and show 95% quantile bands; no statistical significance testing or error bars in an experimental sense — these are illustrative simulations, not benchmarks. No tables in the provided text. Visualization is functional but minimal.

Follow-up references: - Jazwinski (1970) — the classical probabilistic treatment this book explicitly extends; essential for theoretical depth. - Bar-Shalom, Li & Kirubarajan (2001) — comprehensive reference on Kalman filtering and target tracking applications. - Rasmussen & Williams (2006) — Gaussian Process Regression; the book notes connections between GP regression and optimal filtering. - Doucet & Johansen (cited implicitly via SMC framework) — Not stated directly in provided text; Godsill & Rayner (1998) and Andrieu et al. (2002) are cited for particle MCMC extensions.

Third pass — critique

Implicit assumptions: - Markov property of the state sequence is assumed throughout; violation (e.g., long-range dependencies, fractional Brownian dynamics) breaks recursive factorization and all derived algorithms. - For Gaussian approximation methods (EKF, UKF, GHKF, CKF): the posterior is assumed unimodal and well-approximated as Gaussian; multimodal posteriors will cause all Gaussian filters to fail silently with no diagnostic. - Particle filter convergence assumes independent, identically weighted particles after resampling; particle impoverishment in high-dimensional state spaces is noted but no solution is given in the provided text. - Process and measurement noise models are assumed known (or separately estimated); misspecified noise covariances degrade all algorithms but robustness is not analyzed.

Missing context or citations: - No engagement with expectation propagation (EP) or variational Bayes as alternatives to assumed-density filtering, despite listing "EP" in the abbreviations. - Stein's method and Berry–Esseen bounds for Gaussian approximation quality: not cited. - Cubature rules beyond third-degree spherical: not discussed. - Continuous-discrete filtering (SDEs observed at discrete times) is mentioned briefly but not developed despite being common in engineering; connections to Itô calculus and Stratonovich integrals are deferred to "Further topics."

Possible experimental / analytical issues: - No quantitative comparison of EKF vs. UKF vs. GHKF vs. CKF on any test problem; a reader cannot use this book alone to select among them for a given problem — only qualitative guidance is given in the epilogue. - The resonator example used in Chapter 1 for illustration is linear-Gaussian; it therefore cannot demonstrate where approximate nonlinear methods differ from the exact solution. - MCMC and particle MCMC methods (Chapter 12) are introduced without discussing mixing diagnostics, convergence criteria, or computational cost scaling — critical for practical use. - MATLAB code is available on the publisher website but reproducibility depends on external URL persistence; no version pinning or archival repository is mentioned.

Ideas for future work: - Benchmark all covered filter/smoother variants (EKF, UKF, CKF, GHKF, PF with N particles) on common nonlinear problems (e.g., bearings-only tracking, Lorenz-63) with RMSE and wall-clock time to give practitioners a decision matrix. - Extend the general Gaussian smoother framework to continuous-time SDE models (Stratonovich/Itô) and show which discrete-time algorithms are consistent discretizations. - Develop adaptive particle count strategies within the particle EM algorithm to balance estimation accuracy against computational budget as a function of posterior complexity. - Analyze Gaussian filter approximation error as a function of nonlinearity degree and noise magnitude to provide theoretical guidance on when EKF vs. UKF vs. cubature is preferable.