Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods

Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods Konstantin Zuev Institute for Risk and Uncertainty University of Liverpool http://www.liv.ac.uk/risk-and-uncertainty/staff/k-zuev/ 8 Nov, 2013 Workshop on Risk and Uncertainty Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 1 / 11

MCMC Revolution P. Diaconis (2009), The Markov chain Monte Carlo revolution :...asking about applications of Markov chain Monte Carlo (MCMC) is a little like asking about applications of the quadratic formula... you can take any area of science, from hard to social, and find a burgeoning MCMC literature specifically tailored to that area. decoupl i ngbl og. com Konstantin Zuev (University of Liverpool) sl owmuse. wor dpr ess. com sl ow RE and UQ by MCMC wwwt hphys. physi cs. ox. ac. uk comput er weekl y. com Workshop on Risk and Uncertainty 2 / 11

MCMC Revolution P. Diaconis (2009), The Markov chain Monte Carlo revolution :...asking about applications of Markov chain Monte Carlo (MCMC) is a little like asking about applications of the quadratic formula... you can take any area of science, from hard to social, and find a burgeoning MCMC literature specifically tailored to that area. decoupl i ngbl og. com Main message: sl owmuse. wor dpr ess. com sl ow wwwt hphys. physi cs. ox. ac. uk comput er weekl y. com MCMC algorithms can be efficiently used for solving engineering problems involving risk and uncertainty Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 2 / 11

Outline 1 What problems is MCMC meant to solve? 2 Why is MCMC useful in Engineering? 3 How does MCMC work? Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 3 / 11

Purpose of MCMC Problem: To estimate I = h(θ)π(θ)dθ R d parameter space h : R function of interest π(θ) target PDF on Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 4 / 11

Purpose of MCMC Problem: To estimate I = h(θ)π(θ)dθ = E π [h] R d parameter space h : R function of interest π(θ) target PDF on Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 4 / 11

Purpose of MCMC Problem: To estimate I = h(θ)π(θ)dθ = E π [h] R d parameter space h : R function of interest π(θ) target PDF on Easy Cases: d is small (d = 1, 2, 3) numerical integration π(θ) is easy to sample from Monte Carlo method Typical Case in Applications: d is large (d 10 2 10 3 ) π(θ) is known only up to a constant, π(θ) = cf(θ) Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 4 / 11

Purpose of MCMC Problem: To estimate I = h(θ)π(θ)dθ = E π [h] R d parameter space h : R function of interest π(θ) target PDF on Easy Cases: d is small (d = 1, 2, 3) numerical integration π(θ) is easy to sample from Monte Carlo method Typical Case in Applications: d is large (d 10 2 10 3 ) π(θ) is known only up to a constant, π(θ) = cf(θ) Solution: Use an appropriate MCMC method Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 4 / 11

Outline 1 What problems is MCMC meant to solve? 2 Why is MCMC useful in Engineering? Bayesian Inference Performance-Based Design Optimization Reliability Problem 3 How does MCMC work? Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 5 / 11

Bayesian Inference M the assumed model class for the target dynamic system: set of I/O probability models p(y θ, u) u System y θ the uncertain model parameters prior PDF π0(θ) over Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 6 / 11

Bayesian Inference M the assumed model class for the target dynamic system: set of I/O probability models p(y θ, u) u System y θ the uncertain model parameters prior PDF π0(θ) over Bayesian approach: Update π 0 (θ) to posterior PDF π(θ D) via Bayes theorem: π(θ D) = L(D θ)π 0(θ) Z D the measured data from the system L(D θ) the likelihood function Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 6 / 11

Bayesian Inference M the assumed model class for the target dynamic system: set of I/O probability models p(y θ, u) u System y θ the uncertain model parameters prior PDF π0(θ) over Bayesian approach: Update π 0 (θ) to posterior PDF π(θ D) via Bayes theorem: π(θ D) = L(D θ)π 0(θ) Z D the measured data from the system L(D θ) the likelihood function Problems: Posterior expectations E π [h] = h(θ)π(θ D)dθ Evidence Z = L(D θ)π 0 (θ)dθ Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 6 / 11

Performance-Based Design Optimization M the assumed model class for the target dynamic system θ the uncertain model parameters ϕ Φ design variables: controllable parameters that define the system design The objective function for a robust-to-uncertainties design: E π [h(ϕ, θ)] = h(ϕ, θ)π(θ ϕ)dθ h(ϕ, θ) a performance function of the system, e.g. stress in a component π(θ ϕ) a PDF, which incorporates available knowledge about the system Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 7 / 11

Performance-Based Design Optimization M the assumed model class for the target dynamic system θ the uncertain model parameters ϕ Φ design variables: controllable parameters that define the system design The objective function for a robust-to-uncertainties design: E π [h(ϕ, θ)] = h(ϕ, θ)π(θ ϕ)dθ h(ϕ, θ) a performance function of the system, e.g. stress in a component π(θ ϕ) a PDF, which incorporates available knowledge about the system Problem: ϕ = arg min ϕ Φ E π[h(ϕ, θ)] Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 7 / 11

Reliability Problem Reliability Problem: To estimate the probability of failure p F p F = P(θ F ) = π(θ)i F (θ)dθ R d Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 8 / 11

Reliability Problem Reliability Problem: To estimate the probability of failure p F p F = P(θ F ) = π(θ)i F (θ)dθ R d Notation: θ R d represents the uncertain input load θ is a random vector and has joint PDF π F R d a failure domain (unacceptable performance) F = {θ : g(θ) b } g(θ) a performance function b a critical threshold for performance I F (θ) = 1 if θ F and I F (θ) = 0 if θ / F Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 8 / 11

MCMC: The Main Idea Monte Carlo method h(θ)π(θ)dθ 1 N N h(θ i ), i=1 θ i i.i.d π Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 10 / 11

MCMC: The Main Idea Monte Carlo method h(θ)π(θ)dθ 1 N N h(θ i ), i=1 θ i i.i.d π Main difficulty: How to obtain i.i.d. samples from π? Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 10 / 11

MCMC: The Main Idea Monte Carlo method h(θ)π(θ)dθ 1 N N h(θ i ), i=1 θ i i.i.d π Main difficulty: How to obtain i.i.d. samples from π? MCMC samples from π and computes integrals using Markov chains: 1 h(θ)π(θ)dθ N N 0 N h(x i ) i=n 0+1 Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 10 / 11

Papers Applications of MCMC to Engineering Problems: Bayesian Inference J.L. Beck and K.M. Zuev (2013), Asymptotically independent Markov sampling: a new Markov chain Monte Carlo scheme for Bayesian inference, Int. J. for Uncertainty Quant., 3(5), 445-474. Performance-Based Design Optimization K.M. Zuev and J.L. Beck (2013), Global optimization using the asymptotically independent Markov sampling method, Computers & Structures, 126, 107-119. Reliability Problem K.M. Zuev, J.L. Beck, S.K. Au, and L.S. Katafygiotis (2012), Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions, Computers & Structures, 92-93, 283-296. K.M. Zuev and L.S. Katafygiotis (2011), Modified Metropolis-Hastings algorithm with delayed rejection, Probabilistic Engineering Mechanics, 26(3), 405-412. K.M. Zuev and L.S. Katafygiotis (2011), The Horseracing Simulation algorithm for evaluation of small failure probabilities, Probabilistic Engineering Mechanics, 26(2), 157-164. Konstantin Zuev (University of Liverpool) RE and UQ by MCMC Workshop on Risk and Uncertainty 11 / 11