James L. Beck
George W. Housner Professor of Engineering and Applied Science
Department of Computing and Mathematical Sciences
Department of Mechanical and Civil Engineering




Summer Lectures at Caltech on Stochastic System Analysis and Bayesian Updating (2005)



The course syllabus is available for download: syllabus

1. Introduction [Beck]
Big picture: knowledge models, information processing, decision making; general class of stochastic state-space models for dynamical systems; conversion of continuous-time stochastic DE to one-step ahead discrete-time predictive PDF; nominal vs robust analysis; basic problems to be addressed in course.

2. Basic concepts in probability and information [Beck]
Probability as a multi-valued logic for plausible reasoning with incomplete information; some history of its origins: Bayes, Laplace, Jeffreys, Cox, Jaynes; Cox's derivation of probability axioms for statements; Kolmogorov axioms for the special case of sets; probability models for discrete and continuous variables; marginalization and theorem of total probability; irrelevant information and independence; quantifying missing information using probability; Kullback-Liebler relative information; mutual information; information entropy; principle of maximum entropy and exponential family of PDFs.

3. Stochastic predictive analysis theory for single model class [Beck]
Basic ingredients: model class of predictive PDFs with prior PDFs; illustrative examples of probability model classes; initial (prior) predictive analysis; Bayesian model updating: block and sequential modes; updated (posterior) predictive analysis; asymptotic approximations for prior and posterior predictive analysis: identifiability and domain of applicability of parameter estimation (e.g. maximum likelihood); illustrative examples, including linear and nonlinear dynamics in time and frequency domains.

4. Stochastic predictive analysis theory for multiple model classes [Beck]
Prior and posterior predictive analysis with set of model classes (model averaging); Bayesian updating for model class selection and its intrinsic principle of parsimony, with an information-theoretic interpretation; large-sample approximations: Ockham factor and BIC; choice of prior PDFs, including non-informative priors, Jeffrey priors, conjugate priors and maximum entropy priors; illustrative examples of model class selection for dynamical systems.

5. Stochastic system analysis for rare events: approximate analytical methods [Au]
Reliability theory and first-order and second-order approximate methods for calculating failure probabilities for static problems; first-passage probabilities for dynamical systems under stochastic excitation: approximate calculation using Rice's out-crossing theory and application to linear dynamical systems using Liapunov's equation.

6. Stochastic system analysis for rare events: stochastic simulation methods [Au]
Basic Monte Carlo simulation, limiting properties of estimators; simulation of stochastic processes; variance reduction techniques: importance sampling; ISEE method for first-passage probabilities for linear dynamical systems; Markov Chain Monte Carlo methods and Metropolis-Hastings algorithm; Subset Simulation and illustrative example of structural dynamics under seismic risk with failure analysis.

7. Stochastic simulation methods for Bayesian updating [Ching]
Gibbs sampler with adaptive rejection sampling; hybrid Monte Carlo; slice sampling; multi-level Metropolis-Hastings with re-sampling; Bayesian model class selection; illustrative examples for model updating of linear dynamical systems.

8. Bayesian updating for sequential state and parameter estimation [Ching]
Graphical representation of probability models; uncertainty propagation for general systems using moment-matching and maximum entropy; Kalman filter and smoother for linear systems; extended Kalman filter and unscented Kalman filter for nonlinear systems; stochastic simulation methods: particle filter and smoother.

9. Other topics
If time is available other topics will be covered, such as Bayesian linear regression and classification [Ching] and research on reliability-based robust stochastic control [Beck]


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