Lectures on Stochastic System Analysis
and Bayesian Updating
June 29-July 13 2005
James L. Beck, California Institute of Technology
Jianye Ching, National Taiwan University of Science & Technology
Siu-Kui (Ivan) Au, Nanyang Technological University, Singa

Stochastic System Analysis and Bayesian Model Updating
BAYESIAN ANALYSIS: FOREWORDS
Notation
1. System means the real thing and a model is an assumed mathematical form
for the system.
2. The probability model class M contains the set of the all admissible

Stochastic System Analysis and Bayesian Model Updating
BAYESIAN MODEL CLASS SELECTION
Introduction:
We must first ask a question before starting the topic of model class selection:
What constitutes a good model class?
Option 1: A good model class is a set

Stochastic System Analysis and Bayesian Model Updating
BAYESIAN STATE ANALYSIS ON LINEAR GAUSSIAN DYNAMICAL
SYSTEMS
Outline:
Bayesian linear regression
Bayesian state analysis on linear Gaussian dynamical systems
Kalman filter
RTS (Rauch-Tung-Striebel) sm

James L. Beck
5 July 2005
INFORMATION AND PROBABILITY: Part 2
Principle of Maximum Entropy
N
Suppose discrete variable x has the set of possible values X = cfw_ x1 ,., xN = cfw_ xn
n =1
and we want to choose a probability model for x. Let f specify that

Stochastic System Analysis and Bayesian Model Updating
BAYESIAN STATE ANALYSIS ON NONLINEAR DYNAMICAL SYSTEMS
Outline:
Particle filter
Particle smoother
Particle filter
Introduction:
In the case that the state-space model class is nonlinear, Kalman filter

James L Beck
12 July 2005
PROBABILITY LOGIC: Part 2
Axioms for Probability Logic
Based on general considerations, we derived axioms for:
P(b | a) = measure of the plausibility of proposition b conditional on the information
stated in proposition a.
For pr

Model Class Selection
Model
Given: Data D from system and set M of
candidate model classes
M = cfw_M 1 , M 2 ,., M J
where each model class M j defines a set
of possible predictive models for system:
cfw_ p(Yn | Un , j ): j j R
& a probability model p (

Example: Ground Motion Attenuation
Problem: Predict the probability distribution for Peak
Ground Acceleration (PGA), the level of ground shaking
caused by an earthquake
Earthquake records are used to update the predictive probability
for PGA based on eart

Axioms for Probability Logic
P(b | a) = measure of the plausibility of proposition b conditional on the
information stated in proposition a
For propositions a, b and c:
P1: P(b | a) 0
P2: P(b | a & b) = 1
P3: P(b | a) + P(~ b | a) = 1
P4: P(c & b | a) = P

Stochastic System Analysis and Bayesian Model Updating
STOCHASTIC SIMULATION FOR BLOCKED DATA
Monte Carlo simulation
Rejection sampling
Importance sampling
Markov chain Monte Carlo
Monte Carlo simulation
Introduction:
If we know how to directly sample fro