Markov chain Monte Carlo and its Application to some ...zuev/talks/MCMC_app.pdf · Markov chain Monte Carlo and its Application to some Engineering Problems Konstantin Zuev Department

Markov chain Monte Carlo and

its Application to some Engineering Problems

Konstantin Zuev

Department of Computing & Mathematical Sciences

California Institute of Technology

http://www.its.caltech.edu/∼zuev

May 24, 2011

Mechanical & Civil Engineering Seminar, Caltech

Konstantin Zuev (Caltech) MCMC and its Application Mechanical & Civil Engineering Sem. 1 / 32

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.

decouplingblog.comslowslowmuse.wordpress.com www-thphys.physics.ox.ac.uk computerweekly.com

The main goal of this talk: To demonstrate how MCMC algorithms can be used

for solving engineering problems


Outline

1 What problems is MCMC meant to solve?

2 Why is MCMC useful in Engineering?

3 How does MCMC work?

4 MCMC applications to Reliability Problem

5 Summary


Outline





5 Summary


Purpose of MCMC

Problem: To estimate

I =

∫Θ

h(θ)π(θ)dθ

= Eπ[h]

Θ ⊆ Rd parameter space

h : Θ→ R function of interest

π(θ) “target” PDF on Θ

“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 ∼ 102 − 103)

π(θ) is known only up to a constant, π(θ) = cf(θ)

Solution: Use an appropriate MCMC method


Purpose of MCMC


I =

∫Θ

h(θ)π(θ)dθ = Eπ[h]




“Easy” Cases:




d is large (d ∼ 102 − 103)




Purpose of MCMC


I =

∫Θ





“Easy” Cases:




d is large (d ∼ 102 − 103)




Purpose of MCMC


I =

∫Θ





“Easy” Cases:




d is large (d ∼ 102 − 103)




Purpose of MCMC


I =

∫Θ





“Easy” Cases:




d is large (d ∼ 102 − 103)




Outline



I Bayesian Inference

I Optimal Stochastic Design

I Reliability Problem



5 Summary


Bayesian Inference

M the assumed model class for the target dynamic system:

I set of I/O probability models p(y|θ, u)u−→ System

y−→I θ ∈ Θ the uncertain model parameters

I prior PDF π0(θ) over Θ

Bayesian approach:

Update π0(θ) to posterior PDF π(θ|D) via Bayes’ theorem:

π(θ|D) = L(D|θ)π0(θ)/Z

D the measured data from the system

L(D|θ) the likelihood function

Problems:

Evidence

Z =

∫Θ

L(D|θ)π0(θ)dθ

Posterior expectations

Eπ[h] =

∫Θ

h(θ)π(θ|D)dθ


Bayesian Inference





Bayesian approach:





Problems:

Evidence

Z =

∫Θ

L(D|θ)π0(θ)dθ


Eπ[h] =

∫Θ

h(θ)π(θ|D)dθ


Bayesian Inference





Bayesian approach:





Problems:

Evidence

Z =

∫Θ

L(D|θ)π0(θ)dθ


Eπ[h] =

∫Θ

h(θ)π(θ|D)dθ


Optimal Stochastic Design Problem

M the assumed model class for the target dynamic system

θ ∈ 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:

Eπ[h(ϕ, θ)] =

∫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 minϕ∈Φ

Eπ[h(ϕ, θ)]







Eπ[h(ϕ, θ)] =

∫Θ




Problem:


Eπ[h(ϕ, θ)]







Eπ[h(ϕ, θ)] =

∫Θ




Problem:


Eπ[h(ϕ, θ)]


Reliability Problem

Reliability Problem: To estimate the probability of failure pF

pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:

θ ∈ Rd represents the uncertain input load

I θ is a stochastic vector and has joint PDF π

F ⊂ Rd a failure domain (unacceptable performance)

F = {θ : G(θ) ≥ b∗}

G(θ) a performance function

b∗ a critical threshold for performance

IF (θ) = 1 if θ ∈ F and IF (θ) = 0 if θ /∈ F


Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:

θ ∈ Rd represents the uncertain input load

I θ is a stochastic vector and has joint PDF π

F ⊂ Rd a failure domain (unacceptable performance)

F = {θ : G(θ) ≥ b∗}

G(θ) a performance function

b∗ a critical threshold for performance

IF (θ) = 1 if θ ∈ F and IF (θ) = 0 if θ /∈ F


Outline




I Markov chains

I Markov chain Monte Carlo

I Metropolis-Hastings algorithm


5 Summary


MCMC: The Main Idea

Monte Carlo method∫Θ

h(θ)π(θ)dθ ≈ 1

N

N∑i=1

h(θi), θii.i.d∼ π

How to obtain i.i.d. samples from π ?

MCMC samples from π and computes integrals using Markov chains:

∫Θ


N −N0

N∑i=N0+1

h(xi)


MCMC: The Main Idea



N

N∑i=1




∫Θ


N −N0

N∑i=N0+1

h(xi)


MCMC: The Main Idea



N

N∑i=1




∫Θ


N −N0

N∑i=N0+1

h(xi)


Markov chains

K(x,Rd) ≡ 1

K(x, {x}) is not necessarily zero

DefinitionMarkov chain is a sequence of stochastic vectors x0, x1, x2, . . . such that

P (xn+1 ∈ A|x0, . . . , xn) = P (xn+1 ∈ A|xn) ≡ K(xn, A)


Markov chains

K(x,Rd) ≡ 1





Markov chains

K(x,Rd) ≡ 1





Markov chains

K(x,Rd) ≡ 1





Markov chains

Markov chain theory: To describe the distribution of xn when n→∞

DefinitionA probability distribution π is called stationary distribution for K if

π(A) =

∫K(x,A)π(x)dx, for all measurable sets A

The central result: Let K be a transition kernel with a stationary distribution

π where K satisfies certain “ergodic conditions”, then

π is the unique stationary distribution

xn ∼ π, when n→∞


Markov chains



π(A) =







Markov chains



π(A) =







Markov chain Monte Carlo

MCMC methods: The stationary distribution π is given:

it is the target distribution we want to sample from

The main goal: To find an appropriate transition kernel K

K(x, dy) = k(x, y)dy + r(x)δx(dy)

Jumps from x to y 6= x occur

according to k(x, y)

r(x) = 1−∫k(x, y)dy

probability that the chain remains at x

δx(dy) = 1 if x ∈ dy and 0 otherwise

The central result: If k satisfies the detailed balance equation

π(x)k(x, y) = π(y)k(y, x)

then π is the stationary distribution for K









r(x) = 1−∫k(x, y)dy














r(x) = 1−∫k(x, y)dy














r(x) = 1−∫k(x, y)dy














r(x) = 1−∫k(x, y)dy














r(x) = 1−∫k(x, y)dy














r(x) = 1−∫k(x, y)dy














r(x) = 1−∫k(x, y)dy







Metropolis-Hastings algorithm

kMH(x, y) = S(y|x)a(x, y)

S(y|x) proposal distribution

a(x, y) acceptance probability

a(x, y) = min

{1,π(y)S(x|y)

π(x)S(y|x)

}

N. Metropolis et al, 1953

I Statistical mechanics

I Boltzmann distribution

I S(x|y) = S(y|x)

I Absence of Markov chains

W.K. Hastings, 1970

I Generalization of the M-algorithm

I Markov chains come into play

I Non-symmetric proposals

I Different acceptance probabilities






a(x, y) = min

{1,π(y)S(x|y)

π(x)S(y|x)

}




I S(x|y) = S(y|x)


W.K. Hastings, 1970










a(x, y) = min

{1,π(y)S(x|y)

π(x)S(y|x)

}




I S(x|y) = S(y|x)


W.K. Hastings, 1970










a(x, y) = min

{1,π(y)S(x|y)

π(x)S(y|x)

}




I S(x|y) = S(y|x)


W.K. Hastings, 1970






Outline





I Subset Simulation

I Enhancements for Subset Simulation

5 Summary


Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:

θ ∈ Rd is a stochastic vector with the joint PDF π

F ⊂ Rd is a failure domain, F = {θ : G(θ) ≥ b∗}IF (θ) = 1 if θ ∈ F and IF (θ) = 0 if θ /∈ F

Typically in Applications:

d is very large, d ∼ 1000 ⇒ numerical quadrature methods are not suitable

We can calculate IF (θ) for any θ, but it is expensive (dyn. system analysis)

pF is very small, pF ∼ 10−3 − 10−6 ⇒ Monte Carlo method is not suitable

We need MCMC based simulation methods


Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:




d is very large, d ∼ 1000

⇒ numerical quadrature methods are not suitable





Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:









Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:









Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:






pF is very small, pF ∼ 10−3 − 10−6

⇒ Monte Carlo method is not suitable



Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:









Reliability Problem


pF = P (θ ∈ F ) =

∫Rd

π(θ)IF (θ)dθ

Notation:









Subset SimulationS.K. Au & J.L. Beck (PEM, 2001)

Rd = F0 ⊃ F1 ⊃ . . . ⊃ Fm = F

F = {θ : G(θ) ≥ b∗}

Fi = {θ : G(θ) ≥ b∗i }

b∗1 < b∗2 < . . . < b∗m = b∗

pF =

m−1∏k=0

P (Fk+1|Fk)

P (Fk+1|Fk) ≈ 1

N

N∑i=1

IFk+1(θ

(i)k )

θ(i)k ∼ π(θ|Fk) =

π(θ)IFk(θ)

P (Fk)

How to sample from π(·|Fk)?

Use an appropriate MCMC alg



Rd = F0 ⊃ F1 ⊃ . . . ⊃ Fm = F

F = {θ : G(θ) ≥ b∗}

Fi = {θ : G(θ) ≥ b∗i }

b∗1 < b∗2 < . . . < b∗m = b∗

pF =

m−1∏k=0

P (Fk+1|Fk)

P (Fk+1|Fk) ≈ 1

N

N∑i=1

IFk+1(θ

(i)k )


π(θ)IFk(θ)

P (Fk)





Rd = F0 ⊃ F1 ⊃ . . . ⊃ Fm = F

F = {θ : G(θ) ≥ b∗}

Fi = {θ : G(θ) ≥ b∗i }

b∗1 < b∗2 < . . . < b∗m = b∗

pF =

m−1∏k=0

P (Fk+1|Fk)

P (Fk+1|Fk) ≈ 1

N

N∑i=1

IFk+1(θ

(i)k )


π(θ)IFk(θ)

P (Fk)





Rd = F0 ⊃ F1 ⊃ . . . ⊃ Fm = F

F = {θ : G(θ) ≥ b∗}

Fi = {θ : G(θ) ≥ b∗i }

b∗1 < b∗2 < . . . < b∗m = b∗

pF =

m−1∏k=0

P (Fk+1|Fk)

P (Fk+1|Fk) ≈ 1

N

N∑i=1

IFk+1(θ

(i)k )


π(θ)IFk(θ)

P (Fk)





Rd = F0 ⊃ F1 ⊃ . . . ⊃ Fm = F

F = {θ : G(θ) ≥ b∗}

Fi = {θ : G(θ) ≥ b∗i }

b∗1 < b∗2 < . . . < b∗m = b∗

pF =

m−1∏k=0

P (Fk+1|Fk)

P (Fk+1|Fk) ≈ 1

N

N∑i=1

IFk+1(θ

(i)k )


π(θ)IFk(θ)

P (Fk)




Modified Metropolis-Hastings algorithm

S.K. Au & J.L. Beck (PEM, 2001): Standard Metropolis-Hastings algorithm

is not efficient in high dimensions

Generate candidate state y

For each j = 1 . . . N :

I Simulate yj ∼ Sj(·|xjn)

I Compute the acceptance probability

aj(xjn, yj) = min

{1,πj(y

j)

πj(xjn)

}

I Accept/Reject yj

yj =

yj , with prob. aj(xjn, yj)

xjn, with prob. 1− aj(xjn, yj)

Accept/Reject y

xn+1 =

y, if y ∈ Fixn, if y /∈ Fi


Modified Metropolis-Hastings algorithm

S.K. Au & J.L. Beck (PEM, 2001): Standard Metropolis-Hastings algorithm

is not efficient in high dimensionsGenerate candidate state y

For each j = 1 . . . N :

I Simulate yj ∼ Sj(·|xjn)

I Compute the acceptance probability

aj(xjn, yj) = min

{1,πj(y

j)

πj(xjn)

}

I Accept/Reject yj

yj =

yj , with prob. aj(xjn, yj)

xjn, with prob. 1− aj(xjn, yj)

Accept/Reject y

xn+1 =

y, if y ∈ Fixn, if y /∈ Fi


Efficiency of Subset Simulation

Linear Problem

I F ⊂ Rd is a hyperplane

I d = 1000

I pF = 10−k, k = 3, 4, 5, 6

What computational effort is required to achieve the COV δ = 30% ?



Linear Problem


I d = 1000

I pF = 10−k, k = 3, 4, 5, 6




Linear Problem


I d = 1000

I pF = 10−k, k = 3, 4, 5, 6



Outline





I Subset SimulationI Enhancements for Subset Simulation

F Modified Metropolis-Hastings algorithm with Delayed RejectionF Bayesian post-processor for Subset Simulation

5 Summary


Modifications of the Metropolis-Hastings algorithm

Modified MH (MMH) algorithm

I S.K. Au & J.L. Beck, 2001

MH algorithm with delayed rejection (MHDR)

I L. Tierney & A. Mira, 1999

Modified Metropolis-Hastings algorithm with delayed rejection (MMHDR)

I K.M. Zuev & L.S. Katafygiotis, 2011


Metropolis-Hastings algorithm with Delayed Rejection

L. Tierney & A. Mira (Statistics in Medicine, 1999):

a1(xn, y1) = min

{1,π(y1)

π(xn)IF (y1)

}a2(xn, y1, y2) = min

{1,

π(y2)S1(y1|y2)(1− a1(y2, y1))

π(xn)S1(y1|xn)(1− a1(xn, y1))IF (y2)

}

Drawback: Inefficient in high dimensions

Reason: S1(·|xn) and S2(·|xn, y1) are d-dimensional PDFs




a1(xn, y1) = min

{1,π(y1)

π(xn)IF (y1)

}a2(xn, y1, y2) = min

{1,

π(y2)S1(y1|y2)(1− a1(y2, y1))

π(xn)S1(y1|xn)(1− a1(xn, y1))IF (y2)

}






a1(xn, y1) = min

{1,π(y1)

π(xn)IF (y1)

}a2(xn, y1, y2) = min

{1,

π(y2)S1(y1|y2)(1− a1(y2, y1))

π(xn)S1(y1|xn)(1− a1(xn, y1))IF (y2)

}






a1(xn, y1) = min

{1,π(y1)

π(xn)IF (y1)

}a2(xn, y1, y2) = min

{1,

π(y2)S1(y1|y2)(1− a1(y2, y1))

π(xn)S1(y1|xn)(1− a1(xn, y1))IF (y2)

}






a1(xn, y1) = min

{1,π(y1)

π(xn)IF (y1)

}a2(xn, y1, y2) = min

{1,

π(y2)S1(y1|y2)(1− a1(y2, y1))

π(xn)S1(y1|xn)(1− a1(xn, y1))IF (y2)

}




Modified MH algorithm with Delayed Rejection

Features of the Algorithm:

Samples generated by MMHDR are less correlated

then samples generated by MMH.

MMHDR needs more computational effort than MMH

for generating the same number of samples.

Whether MMHDR is useful for reliability problems depends on whether the

gained reduction in variance compensates for the additional cost.

With fixed computational effort:

I MMH: more Markov chains with more correlated states

I MMHDR: fewer Markov chains with less correlated states






























































ExampleLinear problem

Geometry

I d = 1000

I pF = 10−5, β = 4.265

Proposal PDFs

I MMH: Sj(·|xj0) = N (xj0, 1)

I MMHDR: Sj1,2(·|xj0) = N (xj0, 1)

Subset Simulation

MMH(1): SS + MMH, n = 103

MMHDR(1.4): SS + MMHDR, n = 103

MMH(1.4): SS + MMH, n = 1450

Reduction in CV is 11%



Geometry

I d = 1000

I pF = 10−5, β = 4.265

Proposal PDFs

I MMH: Sj(·|xj0) = N (xj0, 1)

I MMHDR: Sj1,2(·|xj0) = N (xj0, 1)

Subset Simulation

MMH(1): SS + MMH, n = 103


MMH(1.4): SS + MMH, n = 1450




Geometry

I d = 1000

I pF = 10−5, β = 4.265

Proposal PDFs

I MMH: Sj(·|xj0) = N (xj0, 1)

I MMHDR: Sj1,2(·|xj0) = N (xj0, 1)

Subset Simulation

MMH(1): SS + MMH, n = 103


MMH(1.4): SS + MMH, n = 1450




Geometry

I d = 1000

I pF = 10−5, β = 4.265

Proposal PDFs

I MMH: Sj(·|xj0) = N (xj0, 1)

I MMHDR: Sj1,2(·|xj0) = N (xj0, 1)

Subset Simulation

MMH(1): SS + MMH, n = 103


MMH(1.4): SS + MMH, n = 1450




Geometry

I d = 1000

I pF = 10−5, β = 4.265

Proposal PDFs

I MMH: Sj(·|xj0) = N (xj0, 1)

I MMHDR: Sj1,2(·|xj0) = N (xj0, 1)

Subset Simulation

MMH(1): SS + MMH, n = 103


MMH(1.4): SS + MMH, n = 1450



“Bayesianization” of Subset Simulation

The key idea of SS:

pF =

m∏k=1

pk, pk = P (Fk|Fk−1)

Original (“frequentist”) SS:

pk ≈ pk =1

N

N∑i=1

IFk(θ

(i)k−1) =

nkN, pF ≈ pSSF =

m∏k=1

nkN

Bayesian SS:

1 Specify prior PDFs p(pk) for all pk = P (Fk|Fk−1), k = 1, . . . ,m.

2 Find the posterior PDFs p(pk|Dk−1) via Bayes’ theorem,

using new data Dk−1 = {θ(1)k−1, . . . , θ

(N)k−1 ∼ π(·|Fk−1)}

3 Obtain the posterior PDF p(pF | ∪m−1k=0 Dk) of pF =

∏mk=1 pk

from p(p1|D0), . . . , p(pm|Dm−1).



The key idea of SS:

pF =

m∏k=1



pk ≈ pk =1

N

N∑i=1

IFk(θ

(i)k−1) =

nkN, pF ≈ pSSF =

m∏k=1

nkN

Bayesian SS:




(N)k−1 ∼ π(·|Fk−1)}


∏mk=1 pk

from p(p1|D0), . . . , p(pm|Dm−1).



The key idea of SS:

pF =

m∏k=1



pk ≈ pk =1

N

N∑i=1

IFk(θ

(i)k−1) =

nkN, pF ≈ pSSF =

m∏k=1

nkN

Bayesian SS:




(N)k−1 ∼ π(·|Fk−1)}


∏mk=1 pk

from p(p1|D0), . . . , p(pm|Dm−1).



The key idea of SS:

pF =

m∏k=1



pk ≈ pk =1

N

N∑i=1

IFk(θ

(i)k−1) =

nkN, pF ≈ pSSF =

m∏k=1

nkN

Bayesian SS:




(N)k−1 ∼ π(·|Fk−1)}


∏mk=1 pk

from p(p1|D0), . . . , p(pm|Dm−1).



The key idea of SS:

pF =

m∏k=1



pk ≈ pk =1

N

N∑i=1

IFk(θ

(i)k−1) =

nkN, pF ≈ pSSF =

m∏k=1

nkN

Bayesian SS:




(N)k−1 ∼ π(·|Fk−1)}


∏mk=1 pk

from p(p1|D0), . . . , p(pm|Dm−1).



The key idea of SS:

pF =

m∏k=1



pk ≈ pk =1

N

N∑i=1

IFk(θ

(i)k−1) =

nkN, pF ≈ pSSF =

m∏k=1

nkN

Bayesian SS:




(N)k−1 ∼ π(·|Fk−1)}


∏mk=1 pk

from p(p1|D0), . . . , p(pm|Dm−1).


Prior and Posterior for pk = P (Fk|Fk−1)

1 Prior PDF p(pk)

Principle of Maximum Entropy:

p(pk) = 1, 0 ≤ pk ≤ 1.

2 Posterior PDF p(pk|Dk−1)

I If θ(1)k−1, . . . , θ

(N)k−1 are i.i.d. according to π(·|Fk−1)

⇒ IFk (θ(1)k−1), . . . , IFk (θ

(N)k−1) can be interpreted as Bernoulli trials

⇒ Bayes’ Theorem (1763):

p(pk|Dk−1) =pnkk (1− pk)N−nk

B(nk + 1, N − nk + 1)

I In fact, θ(1)k−1, . . . , θ

(N)k−1 are MCMC samples (for k ≥ 2)

⇒ θ(1)k−1, . . . , θ

(N)k−1 ∼ π(·|Fk−1), however, they are not independent

p(pk|Dk−1) ≈pnkk (1− pk)N−nk

B(nk + 1, N − nk + 1)



1 Prior PDF p(pk)


p(pk) = 1, 0 ≤ pk ≤ 1.


I If θ(1)k−1, . . . , θ


⇒ IFk (θ(1)k−1), . . . , IFk (θ




B(nk + 1, N − nk + 1)

I In fact, θ(1)k−1, . . . , θ


⇒ θ(1)k−1, . . . , θ



B(nk + 1, N − nk + 1)



1 Prior PDF p(pk)


p(pk) = 1, 0 ≤ pk ≤ 1.


I If θ(1)k−1, . . . , θ


⇒ IFk (θ(1)k−1), . . . , IFk (θ




B(nk + 1, N − nk + 1)

I In fact, θ(1)k−1, . . . , θ


⇒ θ(1)k−1, . . . , θ



B(nk + 1, N − nk + 1)



1 Prior PDF p(pk)


p(pk) = 1, 0 ≤ pk ≤ 1.


I If θ(1)k−1, . . . , θ


⇒ IFk (θ(1)k−1), . . . , IFk (θ




B(nk + 1, N − nk + 1)

I In fact, θ(1)k−1, . . . , θ


⇒ θ(1)k−1, . . . , θ



B(nk + 1, N − nk + 1)



1 Prior PDF p(pk)


p(pk) = 1, 0 ≤ pk ≤ 1.


I If θ(1)k−1, . . . , θ


⇒ IFk (θ(1)k−1), . . . , IFk (θ




B(nk + 1, N − nk + 1)

I In fact, θ(1)k−1, . . . , θ


⇒ θ(1)k−1, . . . , θ



B(nk + 1, N − nk + 1)


Posterior PDF for pF

Last step: To find the PDF of pF =∏mk=1 pk, given the PDFs of all factors

pk ∼ Be(nk + 1, N − nk + 1)

Idea: To approximate pF by a single beta variable

Theorem (Da-Yin Fan, 1991)

Let X1, . . . , Xm be beta variables, Xk ∼ Beta(ak, bk), and Y = X1X2 . . . Xm.

Then Y is approximately distributed as Y ∼ Beta(a, b), where

a = µ1µ1 − µ2

µ2 − µ21

, b = (1− µ1)µ1 − µ2

µ2 − µ21

,

µ1 =

m∏k=1

akak + bk

, µ2 =

m∏k=1

ak(ak + 1)

(ak + bk)(ak + bk + 1).

Nice property of this approximation: E[Y ] = E[Y ], E[Y 2] = E[Y 2]




pk ∼ Be(nk + 1, N − nk + 1)





a = µ1µ1 − µ2

µ2 − µ21

, b = (1− µ1)µ1 − µ2

µ2 − µ21

,

µ1 =

m∏k=1

akak + bk

, µ2 =

m∏k=1

ak(ak + 1)

(ak + bk)(ak + bk + 1).





pk ∼ Be(nk + 1, N − nk + 1)





a = µ1µ1 − µ2

µ2 − µ21

, b = (1− µ1)µ1 − µ2

µ2 − µ21

,

µ1 =

m∏k=1

akak + bk

, µ2 =

m∏k=1

ak(ak + 1)

(ak + bk)(ak + bk + 1).





pk ∼ Be(nk + 1, N − nk + 1)





a = µ1µ1 − µ2

µ2 − µ21

, b = (1− µ1)µ1 − µ2

µ2 − µ21

,

µ1 =

m∏k=1

akak + bk

, µ2 =

m∏k=1

ak(ak + 1)

(ak + bk)(ak + bk + 1).



Bayesian post-processor for Subset Simulation

Point estimate pSSF PDF pSS+(pF ) = Be(pF |a, b)

a =

∏mk=1

nk+1N+2

(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2

b =

(1−

∏mk=1

nk+1N+2

)(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2

What is the relationship between pSS+ and pSSF ?

EpSS+ [pF ]→ pSSF , as N →∞

EpSS+ [pF ] ≈ pSSF , when N is large

The PDF pSS+ can be fully used for life-cost analyses, decision making, etc.

E[Loss(pF )] =

∫Loss(pF )pSS+(pF )dpF




a =

∏mk=1

nk+1N+2

(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2

b =

(1−

∏mk=1

nk+1N+2

)(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2





E[Loss(pF )] =





a =

∏mk=1

nk+1N+2

(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2

b =

(1−

∏mk=1

nk+1N+2

)(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2





E[Loss(pF )] =





a =

∏mk=1

nk+1N+2

(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2

b =

(1−

∏mk=1

nk+1N+2

)(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2





E[Loss(pF )] =





a =

∏mk=1

nk+1N+2

(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2

b =

(1−

∏mk=1

nk+1N+2

)(1−

∏mk=1

nk+2N+3

)∏mk=1

nk+2N+3 −

∏mk=1

nk+1N+2





E[Loss(pF )] =



Elasto-Plastic Structure Subjected to Ground Motion

S.K. Au (Computers & Structures, 2005):

2D moment-resisting steel frame

Synthetic ground motion a = a(Z)

I Z = (Z1, . . . , Zd)i.i.d∼ N (0, 1)

IZ−→ Filter

a(Z)−−−→I d = 1001

Failure domain:

F = {Z ∈ Rd : δmax(Z) > b}

δmax = maxi=1,...,6

δi

δi is the maximum absolute

interstory drift ratio of the ith story

within the duration of study, 30 s

b = 0.5%⇒ pF ≈ 8.9× 10−3


Elasto-Plastic Structure Subjected to Ground Motion

S.K. Au (Computers & Structures, 2005):

2D moment-resisting steel frame

Synthetic ground motion a = a(Z)

I Z = (Z1, . . . , Zd)i.i.d∼ N (0, 1)

IZ−→ Filter

a(Z)−−−→I d = 1001

Failure domain:

F = {Z ∈ Rd : δmax(Z) > b}

δmax = maxi=1,...,6

δi

δi is the maximum absolute

interstory drift ratio of the ith story

within the duration of study, 30 s

b = 0.5%⇒ pF ≈ 8.9× 10−3


Summary

MCMC algorithms computes multi-dimensional integrals using Markov chains

MCMC-based methods can be very useful for solving engineering problems

I Reliability Engineering

Subset Simulation (Au & Beck, 2001)

I a very efficient MCMC method for estimation of small failure probabilities

Enhancements for Subset Simulation

I MMHDR = MMH (Au & Beck, 2001) + MHDR (Tierney & Mira, 1999)

I Bayesian post-processor for Subset Simulation


Summary










Summary










Summary










Acknowledgements

Jim Beck (Caltech)

I for teaching me Bayesian way of thinking

I for mentoring my research at Caltech

Lambros Katafygiotis (HKUST)

I for introducing me to the world of MCMC algorithms

I for supervising me at HKUST

Ivan Au (CityU)

I for providing Matlab code for the nonlinear example

I for being always open for constructive discussions


Thank you for attention!


Documents

Markov chain Monte Carlo and its Application to some ...zuev/talks/MCMC_app.pdf · Markov chain Monte Carlo and its Application to some Engineering Problems Konstantin Zuev Department