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

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

Lectures

Lectures:

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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