reliability engineering and - university of...

Reliability Engineering and System Safety 170 (2018) 146–158

Contents lists available at ScienceDirect

Reliability Engineering and System Safety

journal homepage: www.elsevier.com/locate/ress

System reliability assessment with multilevel information using the

Bayesian melding method

Jian Guo

a , Zhaojun (Steven) Li a , ∗ , Jionghua (Judy) Jin

b

a Department of Industrial Engineering and Engineering Management, Western New England University, Springfield, MA 01119, United States b Industrial and Operations Engineering Department, The University of Michigan, Ann Arbor, MI, 48109, United States

a r t i c l e i n f o

Keywords:

System reliability

Bayesian inference

Bayesian melding method

Multilevel information

Non-informative and informative priors

Markov Chain Monte Carlo

Adaptive sampling importance re-sampling

a b s t r a c t

This paper investigates the Bayesian melding method ( BMM ) for system reliability analysis by effectively integrat-

ing various available sources of expert knowledge and data at both subsystem and system levels. The integration

of multiple priors is investigated under both linear and geometric pooling methods. The aggregated system prior

distributions using various pooling methods including the BMM are evaluated and compared. Based on these

integrated and updated prior distributions and three scenarios of data availability from a system and/or sub-

systems, methods for posterior system reliability inference are proposed. Computational challenges for posterior

inferences using the sophisticated BMM are addressed using the adaptive sampling importance re-sampling ( SIR )

method. A numerical example with simulation results illustrates the applications of the proposed methods and

provides insights for system reliability analysis using multilevel information.

© 2017 Elsevier Ltd. All rights reserved.

1

u

d

i

a

t

I

s

f

b

r

e

t

a

i

s

a

fi

r

u

l

d

w

l

v

t

s

i

f

i

b

B

c

t

a

i

s

w

[

s

g

b

i

m

h

R

A

0

. Introduction

In the product development process, system level reliability testingsually is very costly and could end up with very limited system testata. Domain experts, on the other hand, may have valuable reliabil-ty knowledge based on the understanding about the system such as thedopted new technologies, the complexity of the new product, manufac-uring processes, and other factors that may impact product reliability.ntegrating the limited test data with expert ’s reliability knowledge canignificantly improve the system reliability estimation and reduce costor system level reliability testing and demonstration. Bayesian relia-ility inference has received increasing applications and acceptance foreliability estimation when there are limited data while as abundantxperts and previous knowledge are available [1,2] . For a complex sys-em, there usually exists different levels of reliability prior informationnd test data at system and/or subsystem levels, this brings about greatnterests and challenges in integrating such multilevel information forystem reliability analysis. Traditionally, the system reliability is usu-lly assessed based on its subsystems reliability through the system con-guration and structure function method. However, using the systemeliability information for its subsystem level reliability inferences andpdates has not been well investigated in the literature. With multipleevels of subsystem and system prior information as well as differentata availability, priors aggregation and updating and their integration

∗ Corresponding author.

E-mail address: [email protected] (Z. (Steven) Li).

a

ttps://doi.org/10.1016/j.ress.2017.09.020

eceived 26 November 2016; Received in revised form 2 August 2017; Accepted 30 September

vailable online 20 October 2017

951-8320/© 2017 Elsevier Ltd. All rights reserved.

ith the available data provide many options for posterior system re-iability inference. This paper intends to advance our understanding toarious prior information aggregation methods as well as their integra-ion with available data in system reliability inference. Both system andubsystem priors are assumed to be available considering that a non-nformative prior can always be elicited. Under this assumption, theollowing three scenarios in terms of data availability are investigated,.e., (a) system level data only; (b) subsystem level data only; and (c)oth system and subsystem level data.

When only a single level of data and prior information is available,ayesian reliability inference for a unit, either a subsystem or systeman be proceeded by following the standard Bayesian procedure [3,4] .

In the case that there are multiple priors for the parameters of in-erest, a natural method of prior information integration is to averagell multiple priors [5] . However, the deviations of the different opin-ons are not well quantified in the averaging approach. Pooling methodsuch as the linear pooling and geometric pooling methods allow unequaleights for each prior and emphasize the diversity of multiple priors

6] . In system reliability assessment, system prior can be derived fromubsystem priors through the system configuration and structure. Inte-rating the derived system prior and the natural system prior can alsoe implemented through pooling methods [7,8] . The geometric poolings inspired by the external Bayesianity property [9] . Based on the geo-etric pooling method for multiple priors, Poole et al. [10] proposed Bayesian melding method ( BMM ) to evaluate the population evolu-

2017

https://doi.org/10.1016/j.ress.2017.09.020

http://www.ScienceDirect.com

http://www.elsevier.com/locate/ress

http://crossmark.crossref.org/dialog/?doi=10.1016/j.ress.2017.09.020&domain=pdf

mailto:[email protected]

https://doi.org/10.1016/j.ress.2017.09.020

J. Guo et al. Reliability Engineering and System Safety 170 (2018) 146–158

t

m

s

b

d

u

i

t

(

a

[

i

d

B

f

u

t

B

a

m

u

a

t

f

r

S

p

o

t

n

a

m

c

q

u

v

i

a

u

r

t

e

m

b

d

w

2

2

t

T

𝜋

t

t

sa

h

i

p

d

p

a

𝑇

w

a

o

m

w

𝑇

T

i

l

c

n

a

B

i

p

T

t

u

F

i

e

e

𝑞

𝑞

w

Notation and Acronym

𝜃 The parameters used to model subsystem reliability 𝜙 The parameters used to model system reliability q 1 ( 𝜃) The prior distribution for the parameters in the subsys-

tems reliability model q 2 ( 𝜙) The natural prior distribution of the parameters in the

system reliability model 𝑞 ∗ 1 ( 𝜙) The induced system prior distribution 𝑞 ∗ 𝜙( 𝜙) The pooled system prior distribution

q 𝜙( 𝜙) The updated system prior distribution q 𝜃( 𝜃) The updated subsystem prior distribution M (·) The deterministic function mapping 𝜃 to 𝜙𝜋𝜙( 𝜙) The system posterior distribution 𝜋𝑠𝑢𝑏 𝜃

( 𝜃) The subsystem posterior based on the updated subsystem prior and subsystem level data

𝜋𝑎𝑙𝑙 𝜃( 𝜃) The subsystem posterior based on the updated subsys-

tem prior and both system and subsystem level data BMM Bayesian Melding method

ion of bow-head whales. The geometric pooling-based prior integrationethod of BMM can also update the subsystem priors using the pooled

ystem prior. For Bayesian system reliability assessment, the derived system prior

ased on subsystem priors and the system structure is usually called in-uced system prior, and the existing original system prior is called nat-ral system prior. The integration of the multiple system priors, i.e., thenduced and natural system priors, has been studied to deal with mul-ilevel prior information within a complex system. A natural-conjugateNC) method proposed by Winkler [11] was used to arithmetically aver-ge the induced system prior and the natural system prior. Hamada et al.12] combined multilevel failure information in a fault tree via multiply-ng the high level event and low level event probabilities in the posterioristribution via the full Bayesian theorem. Reese et al. [13] studied theayesian hierarchical method to time-dependent reliability estimationor series systems. In their paper, Metropolis-Hastings ( MH ) method wassed to simulate the posterior distributions. Peng et al. [14] investigatedhe system reliability analysis of multilevel heterozygous data using theayesian theorem. Yontay et al. [15] investigated the system reliabilityssessment using multilevel information through a Bayesian networkethod. Guo et al. [16] estimated the subsystem reliability posteriorssing subsystem level data only through the Bayesian Melding method,nd a traditional “table looking-up ” method is used to estimate the pos-eriors. Wilson et al. [17] summarized Bayesian hierarchical methodsor binary, lifetime, and degradation data. However, the existing systemeliability assessment using BMM considers only a single level of data.ystem and subsystem reliability assessment and updating when multi-le level data are available has not been well investigated. The effectsf using non-informative and informative and various pooling parame-ers on the performance of posteriors under various data availability hasot been studied. Moreover, efficient simulation methods are needed toddress the computational issue when using BMM due to the involvedultilevel prior integration and the complex likelihood function for in-

orporating multiple levels of data. This paper answers these researchuestions in system and subsystem reliability assessment and updatingsing the BMM under the aforementioned three scenarios.

The remainder of the paper is organized as follows. Section 2 re-iews the methods of integrating multiple priors and the Bayesian meld-ng method ( BMM ). Section 3 investigates the BMM for assessing systemnd subsystem reliability based on available data and the integrated andpdated prior distributions ( Fig. 1 ). An adaptive sampling importancee-sampling ( SIR ) method for posterior inference for efficient compu-ation when using the BMM is introduced in this section. A numericalxample is presented to demonstrate the applications of the proposed

147

ethods in Section 4 . The performance and behavior of posterior relia-ility inferences using various pooling parameters under the BMM is alsoiscussed. Section 5 draws the conclusions and discusses future researchork.

. Multiple prior integration

.1. Integration of multiple priors

Let q ( 𝜃), L ( y | 𝜃), and 𝜋( 𝜃| y ) be the prior, likelihood function, and pos-erior distribution of the unknown reliability parameter 𝜃, respectively.he Bayesian posterior reliability can be expressed as

( 𝜃|𝑦 ) ∝ 𝑞( 𝜃) 𝐿 ( 𝑦 |𝜃) . (1)

Expert opinions are one of main sources of the prior distribution forhe parameter 𝜃. An accurate prior will improve the accuracy of pos-erior estimation. Extensive works have been done in prior elicitation,uch as [18] and [19] . With more than one expert, different priors for 𝜃re elicited. It is desirable to integrate these different priors into one co-erent prior probability distribution. There are three methods for pool-ng or averaging multiple priors. One simple method is to average allrior information. The pooling methods can assign unequal weights toifferent priors so that the significance of various priors can be incor-orated. More specifically, the linear pooling stems from the arithmeticverage, which is

( 𝑞 1 , 𝑞 2 , … , 𝑞 𝑘 ) =

𝑘 ∑𝑖 =1

𝛼𝑖 𝑞 𝑖 , (2)

here q i , 𝑖 = 1 , … , 𝑘, are the individual priors, 𝛼i , 𝑖 = 1 , … , 𝑘, ∑𝑘

𝑖 =1 𝛼 = 1 ,re the weights assigned to each corresponding prior. T is the poolingperator that represents the single combined prior distribution. Geo-etric pooling is the product of individual priors powered by unequaleights, which is

( 𝑞 1 , 𝑞 2 , … , 𝑞 𝑘 ) =

𝑘 ∏𝑖 =1

𝑞 𝛼𝑖 𝑖 . (3)

he geometric pooling can be transformed into the form of linear pool-ng by taking logarithm on both sides, which is shown as follow

og 𝑇 ( 𝑞 1 , 𝑞 2 , … , 𝑞 𝑘 ) =

𝑘 ∑𝑖 =1

𝛼𝑖 log 𝑞 𝑖 . (4)

The challenge of coherizing priors linked by a deterministic modelan also be resolved through the pooling methods discussed above. De-ote the deterministic model as 𝜙 = 𝑀( 𝜃) , where 𝜃 and 𝜙 are the inputnd output of this model with priors of q 1 ( 𝜃) and q 2 ( 𝜙), respectively.ased on the prior of q 1 ( 𝜃) and the deterministic model 𝜙 = 𝑀( 𝜃) , an

nduced prior 𝑞 ∗ 1 ( 𝜙) can be assessed, which can be explained as the ex-ert ’s implicit input beliefs about 𝜙 translated through the model M (·).he induced prior can be evaluated as, 𝑞 ∗ 1 ( 𝜙) = 𝑞 1 [ 𝑀

−1 ( 𝜙)] |𝐽 ( 𝜙) | givenhat M (·) is revertible, where | J ( 𝜙)| is the Jacobian matrix of 𝜙. It issually impossible to obtain 𝑞 ∗ 1 ( 𝜙) analytically when M (·) is irreversible.rom the above discussions, we observe that there are two priors, annduced prior 𝑞 ∗ 1 ( 𝜙) and the natural prior q 2 ( 𝜙), about the same param-ter 𝜙. The pooled prior of 𝜙, denoted as 𝑞 ∗

𝜙( 𝜙) , can be evaluated through

ither a linear or geometric pooling algorithm, i.e., Eqs. (5) and (6) ,

∗ 𝜙( 𝜙) ∝ 𝛼𝑞 ∗ 1 ( 𝜙) + (1 − 𝛼) 𝑞 2 ( 𝜙) , (5)

∗ 𝜙( 𝜙) ∝ 𝑞 ∗ 1 ( 𝜙)

𝛼𝑞 2 ( 𝜙) 1− 𝛼, (6)

here 𝛼 is the pooling weight parameter.


System data + priors

Subsystem data+ priors

System & Subsystem data+ priors

System/subsystem posterior

Pooled system prior

Updated subsystem prior+System data+Subsystem data

Updated subsystem posterior

+System structure

Scenario 1 (S1) Scenario 2 (S2) Scenario 3 (S3)

Subsystem prior

Induced system prior

Pooled system prior

+System structure

+System prior

+System dataUpdated subsystem posterior

+ Subsystem data

+System structureUpdated system prior

+System data

+System structureUpdated subsystem prior

Pooled system prior

Update subsystem prior

S11 S12 S2 S3

Fig. 1. Three scenarios for Bayesian system reliability analysis.

2

a

i

I

b

s

b

o

t

n

a

i

t

r

b

s

d

t

𝑀

𝐶

𝑞

r

p

a

o

T

ds

b

t

t

𝑞

𝑞

r

a

i

a

u

w

v

a

𝜙

𝐹

T

𝑞

w

d

c

(

t

c

t

b

s

v

.2. Bayesian melding method (BMM) for prior aggregation and updating

The Bayesian melding method is based on the logarithmic poolinglgorithm of multiple priors linked by a deterministic model as shownn Eq. (6) and intends to update the prior of 𝜃 using the prior of 𝜙.n the system reliability analysis, the deterministic function, denotedy M (·), e.g., the system configuration, defines the mapping from theubsystems ’ parameter set of 𝜃, to the system parameter set of 𝜙. Toe consistent with the existing prior integration methods, the notationsf induced and natural prior distributions are used in BMM . Based onhe logarithmic prior pooling algorithm, a new prior distribution 𝑞 ∗

𝜙( 𝜙) ,

amed the pooled prior distribution of 𝜙 in this paper, can be obtained,s shown in Eq. (6) . When 𝛼 = 0 . 5 , Eq. (6) reduces to a geometric pool-ng by taking the geometric mean of the two prior densities. The deriva-ion process of 𝑞 ∗

𝜙( 𝜙) indicates that the pooled system prior incorpo-

ates the information from both the system and subsystem prior distri-utions of 𝜙 and 𝜃. According to 𝜙 = 𝑀( 𝜃) , 𝑞 ∗

𝜙( 𝜙) can be inverted to the

pace of 𝜃, which yields a new prior distribution, q 𝜃( 𝜃), named the up-ated prior distribution of 𝜃. The derivation of q 𝜃( 𝜃) can be understoodhrough the set theory. Let C 1 be a hyper-cube that contains 𝜃, 𝐶 2 =( 𝐶 1 ) = { 𝜙 = 𝑀( 𝜃) ∶ 𝜃 ∈ 𝐶 1 } , and 𝐶 3 = 𝑀

−1 ( 𝐶 2 ) = { 𝜃 = 𝑀

−1 ( 𝜙) ∶ 𝜙 ∈ 2 } given that M (·) is reversible. Based on the definition of C 3 and 𝑞 ∗ 1 ( ⋅) , 1 ( 𝐶 3 ) = 𝑞 1 [ 𝑀

−1 ( 𝐶 2 )] = 𝑞 ∗ 1 ( 𝐶 2 ) and 𝑞 𝜃( 𝐶 3 ) = 𝑞 𝜃[ 𝑀

−1 ( 𝐶 2 )] = 𝑞 ∗ 𝜙( 𝐶 2 ) . The

atio of q 𝜃( C 1 ) to q 𝜃( C 3 ) is equal to that of q 1 ( C 1 ) to q 1 ( C 3 ) because therior distributions on C 2 provides no information on the relative prob-bility ratio of C 1 and C 3 , which is shown in Eq. (7) for details. Basedn the definitions of C 1 , C 2 , and C 3 , Eq. (7) can be rewritten as Eq. (8) .hus, the updated prior distribution of 𝜃 is equal to the original prioristribution q 1 ( 𝜃) weighted by the ratio of the two densities in the 𝜙pace evaluated by M ( 𝜃), which correspond to the natural prior distri-ution q 2 ( 𝜙) and the induced prior distribution 𝑞 ∗ 1 ( 𝜙) . It is worthwhileo clarify that the prior distribution of 𝜃 updating is not dependent onhe existence of the inverse 𝑀

−1 ( ⋅) of the deterministic function.

𝜃( 𝐶 1 ) = 𝑞 𝜃( 𝐶 3 ) [ 𝑞 1 ( 𝐶 1 ) 𝑞 1 ( 𝐶 3 )

]= 𝑞 ∗

𝜙( 𝐶 2 )

[ 𝑞 1 ( 𝐶 1 ) 𝑞 ∗ 1 ( 𝐶 2 )

]

∝ 𝑞 ∗ 1 ( 𝐶 2 ) 𝛼𝑞 2 ( 𝐶 2 ) 1− 𝛼[ 𝑞 1 ( 𝐶 1 ) 𝑞 ∗ 1 ( 𝐶 2 )

]

= 𝑞 1 ( 𝐶 1 ) [ 𝑞 2 ( 𝐶 2 ) 𝑞 ∗ ( 𝐶 2 )

]1− 𝛼, (7)

1

148

𝜃( 𝜃) ∝ 𝑞 1 ( 𝜃) [ 𝑞 2 [ 𝑀( 𝜃)] 𝑞 ∗ 1 [ 𝑀( 𝜃)]

]1− 𝛼. (8)

With the updated subsystem priors, a new system prior can be de-ived through the system configuration. This new system prior is nameds the updated system prior and denoted as q 𝜙( 𝜙) in this paper. If M (·)s reversible, the updated system prior distribution can be derived in similar way as evaluating the induced system prior of 𝑞 ∗ 1 ( 𝜙) using thepdated subsystem prior distributions q 𝜃( 𝜃), i.e., 𝑞 𝜙 = 𝑞 𝜃[ 𝑀

−1 ( 𝜙)] |𝐽 ( 𝜙) |,here | J ( 𝜙)| is the Jacobin function of 𝜙. Under the condition of a re-ersible M (·) function, the updated system prior and pooled system priorre different to a constant only, which is proved in Eqs. (9) and (10) .

Denote F (·) as the cumulative density function (CDF) of the updatedand the CDF of the updated system prior can be derived as,

( 𝜙) = 𝑃 (Φ < 𝜙) = 𝑃 ( 𝑀( 𝜃) < 𝜙) = 𝑃 ( 𝜃 < 𝑀

−1 ( 𝜙))

= ∫𝑀

−1 ( 𝜙)

0 𝑞 𝜃( 𝜃) 𝑑𝜃 = ∫

𝜙

0 𝑞 𝜃( 𝑀

−1 ( 𝜙)) 𝑑𝑀

−1 ( 𝜙)

= ∫𝜙

0 𝑞 𝜃[ 𝑀

−1 ( 𝜙)] |𝐽 ( 𝜙) |𝑑𝜙= ∫

𝜙

0 𝑘𝑞 1 [ 𝑀

−1 ( 𝜙)] [ 𝑞 2 ( 𝜙) 𝑞 ∗ 1 ( 𝜙)

]1− 𝛼|𝐽 ( 𝜙) |𝑑𝜙. (9)

he pdf of the updated system prior for 𝜙 can be derived as follow,

𝜙( 𝜙) = 𝐹 ′( 𝜙) = 𝑘𝑞 1 [ 𝑀

−1 ( 𝜙)] [ 𝑞 2 ( 𝜙) 𝑞 ∗ 1 ( 𝜙)

]1− 𝛼|𝐽 ( 𝜙) | = 𝑘𝑞 ∗ 1 ( 𝜙) [ 𝑞 2 ( 𝜙) 𝑞 ∗ 1 ( 𝜙)

]1− 𝛼∝ 𝑞 ∗ 1 ( 𝜙)

𝛼𝑞 2 ( 𝜙) 1− 𝛼, (10)

here k is a constant. This concludes that the updated system prior isifferent from the pooled system prior only to a constant.

However, for a given system, the M (·) function describing the systemonfiguration is usually irreversible, and we can not use Eqs. (9) and10) to evaluate the updated system prior. In addition, the pooled sys-em prior and updated system prior will not be only different than aonstant as proved in the following. Without loss of generality, take thewo-component series system for example. Let 𝜃 = ( 𝜃1 , 𝜃2 ) 𝑇 , 𝜃1 , 𝜃2 ∈ [0 , 1]e the reliability parameter of the components in the two-componentystem and 𝑀( 𝜃) = 𝜃1 𝜃2 describe the system figuration, which is irre-ersible. The joint distribution of 𝜃1 and 𝜃2 of 𝑓 1 , 2 = 𝑓 1 𝑓 2 where f 1 and


Fig. 2. The schematic of iterative updating based on BMM.

f

𝑓

𝑓

w

𝑞

e

d

j

e

𝑓

p

s

t

i

s

b

o

i

c

f

p

𝑀

t

o

o

c

t

W

g

t

S

i

𝑏

fi

b

Table 1

Expert estimates on the reliability of each subsystem.

Subsystem Mode ( p e ) Confidence significance ( C e )

Component 1 0.90 10

Component 2 0.90 10

System 0.90 10

d

o

i

t

m

c

u

u

a

s

I

a

m

2

m

p

n

s

t

c

p

p

d

s

s

s

e

t

p

s

f

p

s

e

i

T

p

s

e

g

w

o

n

t

a

t

3

m

a

2 are the updated prior density functions of 𝜃1 and 𝜃2 , respectively,

1 = 𝑘 1 𝑞 1 1 ( 𝜃1 )

[ 𝑞 2 [ 𝑀( 𝜃)] 𝑞 ∗ 1 [ 𝑀( 𝜃)]

]1− 𝛼,

2 = 𝑘 2 𝑞 2 1 ( 𝜃2 )

[ 𝑞 2 [ 𝑀( 𝜃)] 𝑞 ∗ 1 [ 𝑀( 𝜃)]

]1− 𝛼, (11)

here 𝑞 1 1 and 𝑞 2 1 are the original (natural) prior density function, 𝑞 1 ( 𝜃) =

1 1 ( 𝜃1 ) 𝑞

2 1 ( 𝜃2 ) , and k 1 and k 2 are constants. The updated prior of 𝜙 can be

valuated as follows. Given random variables of X, Y and 𝑉 = 𝑋𝑌 , the

ensity function of V is 𝑓 𝑉 ( 𝑣 ) = ∫ +∞−∞ 𝑓 𝑋,𝑌 ( 𝑥,

𝑣

𝑥 ) 1 |𝑥 |𝑑𝑥, where f X, Y is the

oint density function of X and Y [20] . The updated system prior can bevaluated as,

𝜙 = ∫1

0 𝑓 1 , 2

(

𝜃1 , 𝜙

𝜃1

)

1 𝜃1

𝑑𝜃1

= ∫1

0 𝑓 1 ( 𝜃1 ) 𝑓 2

(

𝜙

𝜃1

)

1 𝜃1

𝑑𝜃1

∝[ 𝑞 2 ( 𝜙) 𝑞 ∗ 1 ( 𝜙)

]1− 𝛼∫

1

0 𝑞 1 1 ( 𝜃1 ) 𝑞

2 1

(

𝜙

𝜃1

)

1 𝜃1

𝑑𝜃1 . (12)

By comparing Eqs. (10) and (12) , if we can prove that the inducedrior 𝑞 ∗ 1 ( 𝜙) is not different to the second term in Eq. (12) only to a con-tant, we can conclude that the pooled system prior is not different tohe updated system prior only to a constant when the M (·) function isrreversible. Under the assumption of two independent components ineries, the induced system prior of 𝜙 can be expressed as the joint distri-ution of the two components ’ prior density integrating over all values

f 𝜃1 , i.e., ∫ 1 0 𝑞

1 1 ( 𝜃1 ) 𝑞

2 1 (

𝜙

𝜃1 ) 𝑞 1 1 ( 𝜃1 ) 𝑑𝜃1 . Comparing this expression with the

ntegral in Eq. (12) , it is clear that 𝑞 1 1 ( 𝜃1 ) is not always equal to 1 𝜃1

. This

ompletes our proof that the pooled system prior is not only differentrom the updated system prior to a constant. Likewise, for other com-lex system structures such as the two-component parallel system with( 𝜃) = 𝜃1 + 𝜃2 − 𝜃1 𝜃2 and other mixed parallel and series system struc-

ures, the pooled and updated system prior will be different from eachther not just to a constant.

According to the Bayesian theorem, both subsystem and system pri-rs can be continuously updated via BMM. The iterative updating pro-edure is illustrated in Fig. 2 , where k , indicates the number of itera-ions. 𝑞 (0)

𝜃( 𝜃) and 𝑞 (0)

𝜙( 𝜙) are initialized as q 1 ( 𝜃) and q 2 ( 𝜙), respectively.

ithout loss of generality, a two-component series system is investi-ated as an example to illustrate this iterative updating process. Assumehe prior reliability of subsystems and system are Beta distributions.uper-parameters in priors are given in terms of mode reliability andts confidence level, see Table 1 . That is, for Beta ( a, b ), 𝑎 = 𝐶 𝑝 𝑝 𝑒 + 1 and = 𝐶 𝑝 (1 − 𝑝 𝑒 ) + 1 , where p e and C p are the mode reliability and its con-dence level, respectively. For example, the system reliability prior iseta (10, 2) based on Table 1 . Since the induced prior and pooled prior

149

o not have closed forms, numerical simulation methods are used tobtain the priors. For the purpose of comparison, the case with an non-nformative system prior beta (2, 2) is also investigated. It is noted thathe updated system and subsystem priors skew to the left and spread outore in both cases as the iteration number k increases ( Fig. 3 ). We can

onclude that iterative updating neutralizes the effects of the system nat-ral prior and subsystem priors on the updated system prior. Iterativesing of the system structure could lead to wider CIs due to the vari-tion propagation. Fig. 4 illustrates the convergence trend of updatedystem and subsystem prior with the natural system prior beta (10, 2).t is noted that the updated system and subsystem priors barely changefter 𝑘 = 4250 iterations which is determined through the binary searchethod based on the distribution mode.

.3. Comparison of various system priors with different 𝛼

It is noted that the induced system prior represents the prior infor-ation from subsystems only and does not include the natural systemrior information. The pooled system prior distribution include bothatural system prior and induced system prior information. The updatedystem prior integrates the updated subsystem priors through the sys-em structure function. The example of reliability assessment for a two-omponent serial system is used to compare the natural prior, inducedrior and pooled prior with different pooling parameters. Fig. 5 com-ares the system natural prior distribution, the induced system prioristribution, the pooled system prior distribution as well as the updatedystem prior distribution under various pooling parameters. It is ob-erved that both pooled and updated system priors balance the naturalystem prior and the subsystem prior information. The pooling param-ter determines the weights of the induced prior and natural prior. Inhis example, as 𝛼 increases, the pooled and updated system prior ap-roaches to the induced prior, meaning that more information from theubsystem priors is incorporated in the pooled prior distribution.

According to Bayesian theorem, both prior and likelihood have ef-ects on the posterior. Under a given data set, various levels of availablerior information can lead to different posterior inferences. In this re-earch, we focus on how non-informative and informative prior knowl-dge from both the subsystems and system can influence the posteriornference in conjunction with different scenarios of data availability.he effects to posterior distributions due to various combinations ofrior and data availability are compared by assuming a neutral trueubsystem and system reliability. The posterior distribution and infer-nce differences are further complicated due to the adopted prior ag-regation methods, e.g., pooled system prior and updated system prior,hich are also investigated in this paper. Specifically, four cases in termsf the availability of system and subsystem prior information are: 1)on-informative system and subsystem priors, 2) non-informative sys-em prior and informative subsystem prior, 3) informative system priornd non-informative subsystem prior, 4) informative system and subsys-em priors.

. Posterior inference using the Bayesian melding method

This section discusses the three scenarios for system reliability assess-ent and updating using the BMM under various availability of system

nd subsystem data.


0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

updated system prior with sys.prior beta(2,2)

Reliability

Den

sity

Updated sys.prior, k=1Updated sys. prior, k=50Updated sys. prior, k=1000Updated sys. prior, k=10000

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

67

updated subsystem prior with sys.prior beta(2,2)

Reliability

Den

sity

Updated subsys.prior, k=1Updated subsys. prior, k=50Updated subsys. prior, k=1000Updated subsys. prior, k=10000

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

updated system prior with sys. prior beta(10, 2)

Reliability

Den

sity

Updated sys.prior, k=1Updated sys. prior, k=50Updated sys. prior, k=1000Updated sys. prior, k=10000

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

67

updated subsystem prior with sys. prior beta(10, 2)

Reliability

Den

sity

Updated subsys.prior, k=1Updated subsys. prior, k=50Updated subsys. prior, k=1000Updated subsys. prior, k=10000

Fig. 3. Iterative updating of system and subsystem reliability prior using BMM.

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

Updated system prior

Reliability

Den

sity

k=4250 Mode: 0.323 , CI: 0.693

k=10000 Mode: 0.323 , CI: 0.69

k=20000 Mode: 0.326 , CI: 0.693

updated sys.prior, k=4250updated sys.prior, k=10000updated sys.prior, k=20000

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

67

Updated subsystem prior

Reliability

Den

sity

k=4250 Mode: 0.583 , CI: 0.552

k=10000 Mode: 0.586 , CI: 0.551

k=20000 Mode: 0.584 , CI: 0.552

updated subsys.prior, k=4250updated subsys.prior, k=10000updated subsys.prior, k=20000

Fig. 4. System and subsystem priors convergence after iterative BMM updating.

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

pooling parameter: 0.1

Reliability

Den

sity

natural priorinduced priorpooled priorupdated prior

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4


Reliability

Den

sity


0.0 0.2 0.4 0.6 0.8 1.0

01

23

4


Reliability

Den

sity


Fig. 5. The natural, induced, pooled, and updated system priors with various pooling parameters.

3

a

a

i

T

3

r

F

t

p

p

a

b

𝜋

w

p

l

.1. Posteriors inference based on the Bayesian melding method

There are three scenarios in terms of various availability of systemnd subsystem level data. In Scenario 1 ( S 1), only system level data arevailable, in Scenario 2 ( S 2) only subsystem level data are available andn Scenario ( S 3), both system and subsystem level data are available.his section investigates the posterior inference using the BMM .

.1.1. Scenario 1 (S1): posterior inference using the system level data only

In the case of the existence of system level data only, system poste-ior distribution can be evaluated using two methods, see Scenario 1 in

150

ig. 1 . The first method ( S 11) is to integrate the pooled system prior dis-ribution with the likelihood function using the system level data. Theooled system prior distribution can be evaluated as in Eq. (6) with aooling parameter 𝛼. Based on the pooled system prior and the avail-ble system level data, the system posterior reliability distribution cane evaluated as follows,

𝜙( 𝜙) ∝ 𝑞 ∗ 𝜙( 𝜙) 𝐿 2 ( 𝜙) , (13)

here 𝜋𝜙( 𝜙) represents the system posterior distribution based on theooled system prior distribution and the system level data, L 2 ( 𝜙) is theikelihood function using the system level data.


p

t

l

S

u

t

a

3

t

a

d

p

𝜋

w

s

l

t

t

3

s

t

t

c

s

b

f

m

T

𝜙

a

o

d

f

d

i

s

d

t

s

a

𝜋

3

i

l

f

o

c

B

s

r

a

g

b

O

[

s

c

a

𝜋

t

p

f

p

m

i

s

b

e

p

s

d

r

{

𝑤

𝑤

H

c

w

s

m

c

t

s

p

I

t

s

t

e

w

f

weights obtained from step 3.

The second method ( S 12) is to first derive the updated subsystemriors using the BMM and then to evaluate the updated system priorhrough the system structure. The updated system prior and the systemevel data can then be utilized to assess the posterior system reliability.pecifically, the updated subsystem prior distribution can be evaluatedsing Eq. (8) . Based on the updated system prior and the available sys-em test data, the system reliability posterior distribution can be evalu-ted as,

𝜋𝜙( 𝜙) ∝ 𝑞 𝜙( 𝜙) 𝐿 2 ( 𝜙) , (14)

.1.2. Scenario 2 (S2): posterior inference using the subsystem data only

When only the subsystem level data are available, the subsystem pos-erior can be assessed based on the updated subsystem prior distributions shown in Eq. (8) and the likelihood function for these subsystem levelata. The posterior distribution of 𝜃 based on the updated subsystemrior distribution and subsystem level data can be evaluated as,

𝑠𝑢𝑏 𝜃

∝ 𝑞 𝜃( 𝜃) 𝐿 1 ( 𝜃) ∝ 𝑞 1 ( 𝜃) ( 𝑞 2 [ 𝑀( 𝜃)] 𝑞 ∗ 1 [ 𝑀( 𝜃)]

)1− 𝛼𝐿 1 ( 𝜃) , (15)

here 𝜋𝑠𝑢𝑏 𝜃

( 𝜃) is the posterior distribution of 𝜃 based on the updatedubsystem prior distribution and subsystem level test data, L 1 ( 𝜃) is theikelihood function for subsystem level data. The system reliability pos-erior distribution can be further derived from 𝜋𝑠𝑢𝑏

𝜃( 𝜃) through the sys-

em structure.

.1.3. Scenario 3 (S3): posterior inference using both the system and

ubsystem data

When both system and subsystem level prior and data are available,he BMM provides a way to integrate system level data into the subsys-em posterior distribution. The BM method eliminates the Borel paradoxaused by the Bayesian synthesis method through reformulation via atandard Bayesian procedure [21,22] . We can assume the independenceetween the prior information for 𝜃 and 𝜙 as well as for the likelihoodunctions by considering that multiple independent sources of infor-ation and data are available for the priors and likelihood functions.hen the joint posterior distribution of 𝜃 and 𝜙 can be expressed as p ( 𝜃,

) ∝q 1 ( 𝜃) q 2 ( 𝜙) L 1 ( 𝜃) L 2 ( 𝜙). When considering both induced system priornd natural system prior distributions, there are two prior distributionsn the same quantity. Assuming these two sources of information are in-ependent, and they can be incorporated into a single prior distributionor 𝜃 through the deterministic function 𝜙 = 𝑀( 𝜃) , an updated posterioristribution on 𝜃 can be evaluated as 𝜋𝑎𝑙𝑙

𝜃∝ 𝑞 𝜃( 𝜃) 𝐿 1 ( 𝜃) 𝐿 2 [ 𝑀( 𝜃)] , which

ncorporates both natural and induced priors and likelihood of 𝜃 and 𝜙,ee Eq. (16) . 𝜋𝑎𝑙𝑙

𝜃is the subsystem posterior distribution based on the up-

ated subsystem prior distribution and both system and subsystem levelest data. The system reliability posterior distribution can then be as-essed using the updated subsystem posterior distribution 𝜋𝑎𝑙𝑙

𝜃( 𝜃) under

given system structure.

𝑎𝑙𝑙 𝜃

∝ 𝑞 𝜃( 𝜃) 𝐿 1 ( 𝜃) 𝐿 2 [ 𝑀( 𝜃)] ∝ 𝑞 1 ( 𝜃) ( 𝑞 2 [ 𝑀( 𝜃)] 𝑞 ∗ 1 [ 𝑀( 𝜃)]

)1− 𝛼𝐿 1 ( 𝜃) 𝐿 2 [ 𝑀( 𝜃)] . (16)

.2. Simulation algorithms for posterior inference using BMM

The posterior system and subsystem reliability inferences describedn Section 3.1 involve complex prior distributions, pooled prior, andikelihood functions. This section discussed using simulation techniquesor approximating the posterior probability distributions. MCMC meth-ds, such as Gibbs sampling and Metropolis-Hastings method, are mostommonly used simulation techniques in Bayesian inference [15,23] .ased on the characteristic of the BMM, sampling Importance Re-ampling ( SIR ) initially developed by Rubin [24] is suggested for poste-ior inference to address the challenges of sampling from complex priorsnd likelihood functions. Taddy et al. [25] also suggested to use SIR in

151

enerating samples from the induced and pooled system prior distri-utions. The SIR method has proved computational efficiency which is ( n ) compared with O ( n log n ) when using the standard MCMC method26] .

In SIR method, to make inference on the posterior 𝜋( 𝜃| y ), randomamples are generated from 𝜋( 𝜃| y ). With no closed form, it is diffi-ult to sample from the posterior. The prior q ( 𝜃) is a good choices the proposal distribution. The weight function is then given by( 𝜃|𝑦 )∕ 𝑞 ( 𝜃) = 𝐾𝐿 ( 𝑦 |𝜃) , where L ( y | 𝜃) and K are the likelihood func-ion and the marginal likelihood, respectively. Therefore, with sam-les { 𝜃1 , 𝜃2 , … , 𝜃𝑘 } generated from q ( 𝜃), an approximation samplerom the posterior distribution can be obtained by drawing a sam-le { 𝜃∗ 1 , 𝜃

∗ 2 , … , 𝜃∗

𝑘 } with replacement from { 𝜃1 , 𝜃2 , … , 𝜃𝑛 } with the nor-

alized weight { 𝐿 ( 𝜃1 )∕ ∑

𝐿 ( 𝜃𝑖 ) , 𝐿 ( 𝜃2 )∕ ∑

𝐿 ( 𝜃𝑖 ) , … , 𝐿 ( 𝜃𝑘 )∕ ∑

𝐿 ( 𝜃𝑖 )} . Sim-larly, the standard SIR procedure for Bayesian Melding method can beummarized as follow. Samples of 𝜃 = { 𝜃1 , … , 𝜃𝑛 } , where n is the num-er of samples, are firstly generated from its prior distribution q 1 ( 𝜃). Forach sample, 𝜃i , 𝑖 = 1 , 2 , … , 𝑛, the corresponding system level reliabilityarameter 𝜙 = 𝑀( 𝜃) is evaluated. The sampling importance weight forample 𝜃i is calculated using Eqs. (17) and (18) based on the posterioristribution shown in Eqs. (17) and (18) , respectively. Then the poste-ior distribution of 𝜃 is approximated by the SIR from the initial samples 𝜃1 , … , 𝜃𝑛 } .

𝑖 =

( 𝑞 2 [ 𝑀( 𝜃𝑖 )] 𝑞 ∗ 1 [ 𝑀( 𝜃𝑖 )]

)1− 𝛼𝐿 1 ( 𝜃𝑖 ) (17)

𝑖 =

( 𝑞 2 [ 𝑀( 𝜃𝑖 )] 𝑞 ∗ 1 [ 𝑀( 𝜃𝑖 )]

)1− 𝛼𝐿 1 ( 𝜃𝑖 ) 𝐿 2 [ 𝑀( 𝜃𝑖 )] (18)

owever, the SIR algorithm may not work well due to the issue ofentered large importance weights which occurs occasionally. In otherords, a small number of large importance weights can dominate the re-

ampling process. Liang [27] extended the pruned-enriched Rosenbluthethod by imposing upper bounds on the weights to limit the usage of

ertain samples. The enrichment method distributes samples with ex-remely large weights evenly to the samples with smaller weights. Byplitting large weights into smaller ones, a wider range of samples is ex-lored and the importance re-sampling estimator is still unbiased [28] .n the BMM for the system and subsystem reliability assessment usinghe adaptive SIR algorithm, the weights are the functions of the inducedystem prior, natural system prior, and the likelihood functions of bothhe system and/or subsystems, see Eqs. (17) and (18) . One key param-ter of the adaptive enriched sampling scheme is to set the thresholdeight, which needs to be pre-determined. The adaptive SIR procedure

or Bayesian Melding method is summarized as follows:

1. Set up the number of iteration, denoted as n . 2. For 𝑖 = 1 , … , 𝑛 :

(a) Generate 𝜃i from q 1 ( 𝜃); (b) Evaluate M ( 𝜃i ) based on the sample 𝜃i ;

3. Calculate the re-sampling importance weights, w i , using Eqs. (17) or(18) . (a) Select m weights among the importance weights w i randomly and

sort them in a descending order; (b) Track the cumulative sum of the sorted weights, and assign the

threshold weight 𝑤 𝑟 =

∑⌊𝛾×𝑚 ⌋𝑖 =1 𝑤 𝑖 , where 𝛾 is denoted as an em-

pirical threshold percentage, such as 𝛾 = 90% . ⌊𝛾 ×m ⌋ is the floorvalue of 𝛾 ×m ;

(c) Examine all importance weights and split any weight w j > w r into𝑘 𝑗 = ⌊𝑤 𝑗 ∕ 𝑤 𝑟 + 1 ⌋ number of weights w j / k j for re-sampling thecorresponding 𝜃j ;

(d) Repeat (a)-(c) until the pre-determined condition, such as a cer-tain standard deviation of sampling weights, is reached.

4. Re-sample from initial samples with the sampling importance


System reliability assessment

Data availability

Assess the system posterior using the pooled prior via MCMC

[Equation (14)]

Evaluate the pooled prior using induced and natural prior [Equation (6)]

Evaluate the updated system prior using the updated subsystem prior and system

structure

Evaluate the updated subsystem prior using the pooled system prior

[Equation (8)]

S1: Only system level data available

Evaluate the system posterior using the pooled prior via MCMC

[Equation (13)]

Prior choice

Pooled prior Updated prior

S3: Both system and subsystem level data available

S2: Only subsystem level data available

Evaluate the pooled prior using the induced and natural prior

[Equation (6)]


[Equation (8)]

Assess the subsystem posterior using the updated subsystem prior via adaptive SIR

[Equation (15)]

Assess the system posterior using the subsystem posterior and system structure

Evaluate the pooled prior using induced and natural prior [Equation (6)]

Assess the subsystem posterior using the updated subsystem prior via adaptive SIR

[Equation (16)]


[Equation (8)]

Assess the system posterior using the subsystem posterior and system structure

Fig. 6. The general procedure of system/subsystem reliability assessment via BMM.

Componet1 Component 2

Component 4a

Component 4b

Component 5

Fig. 7. Reliability block diagram (RBD) of a partial missile system.

T

p

o

p

p

s

t

v

4

4

s

s

g

w

t

𝜃

t

𝜙

I

d

r

f

u

a

a

p

Table 2

Expert estimates on the reliability of each subsystem.

Subsystem Mode p e Confidence significance C p

Subsystem 1 0.999 10


Subsystem 4a 0.930 10

Subsystem 4b 0.99 10


r

p

t

p

f

4

t

d

o

w

f

d

o

F

u

c

s

s

i

r

w

he SIR method can simulate and evaluate both subsystem and systemosterior more efficiently by using the samples from the subsystem pri-rs q 1 ( 𝜃) to evaluate the induced system prior and the pooled systemrior distributions. Moreover, the SIR method can avoid repeated sam-ling from each of the prior components for posterior evaluations.

The general procedure of system/subsystem reliability posterior as-essment via Bayesian melding method is illustrated in Fig. 6 . Accordingo the data availability, the system/subsystem priors are estimated witharious priors.

. Numerical example

.1. Case background

The numerical example extends the work by Li and Mense [29] . Theystem of interest is about a critical portion of a missile system whoseubsystems are independent from each other. The reliability block dia-ram is shown in Fig. 7 . Subsystems 1, 2, and 5 are connected in series,hile subsystems 4a and 4b are connected in parallel. Therefore, the de-

erministic function can be described using Eq. (10) , where, 𝜃1 , 𝜃2 , 𝜃5 ,

4 a and 𝜃4 b are the parameters in Bernoulli trials of each subsystem, i.e.,he reliability of subsystems.

= 𝑀( 𝜃) = 𝜃1 𝜃2 𝜃5 ( 𝜃4 𝑎 + 𝜃4 𝑏 − 𝜃4 𝑎 𝜃4 𝑏 ) . (19)

n this application, it is very costly to implement testing for reliabilityemonstration, especially at the system level. Only subsystem level testesults and limited system level historical data in terms of either pass orail are available. The objective is to estimate the parameter 𝜙 and 𝜃 bytilizing limited binary test data and expert prior knowledge.

The non-informative natural system and subsystem prior distributionre given as Beta (2, 2), which represents the prior reliability centersround 0.5, a 50%–50% fail versus pass probability. The informativerior distribution of system and subsystems are given in terms of mode

152

eliability and its confidence level i.e., the parameters for subsystemriors can be estimated as follows, 𝑎 = 𝐶 𝑝 𝑝 𝑒 + 1 and 𝑏 = 𝐶 𝑝 (1 − 𝑝 𝑒 ) + 1 . Inhis research, an expert team was interviewed to provide the informativerior information about the reliability of system and subsystems in theorm of the reliability mode estimation and confidence levels Table 2 .

.2. Simulation method for posterior inference

According to the Bayesian theorem, both prior and likelihood func-ion affect the posterior inferences. To study the effects of priors, theata for the likelihood function are simulated. In this case, pass/fail dataf subsystems and system are generated through binomial distributionsith the certain success probability, such as 0.96 or 0.50, so that the ef-

ects of different pooling methods and pooling parameters on posterioristribution can be well illustrated in the term of modes. To ensure tobtain the rare events, sample size of with simulation runs is selected.or each generated sample data set, a posterior distribution can be eval-ated. The posterior reliability under various pooling mechanisms areompared in terms of the modes in this paper. The effects of the sampleize of system and subsystem data is also considered by using variousample sizes, e.g., 5, 10, 100, and 1000, for posterior system reliabil-ty inferences. Under a given sample size, a large number of simulationuns, e.g., 10,000, can be conducted through the sampling distributionith the assumed true reliability. For the comparison purpose, the mode


Fig. 8. The simulation procedure for the posterior inference.

r

t

F

s

t

4

d

f

u

p

t

4

a

r

S

t

t

i

w

p

b

T

s

T

u

w

o

t

s

t

p

4

s

1

s

d

i

c

B

b

a

p

t

t

a

a

r

r

p

p

d

D

e

t

t

i

h

t

p

p

t

t

c

f

r

c

s

s

a

d

p

a

s

1

j

s

r

m

t

s

h

r

m

t

m

C

i

t

b

t

u

t

b

eliability and its 95% credible interval are used to characterize the pos-erior system reliability. The simulation procedure can be summarizedig. 8 , where rbinom ( n, s, p ) is the function generating test data with n,

, and p as the number of simulation runs, sample size, and the assumedrue reliability, respectively.

.3. System and subsystem reliability assessment using multilevel prior and

ata

In this section, the focus is to integrate system level reliability in-ormation with subsystem level reliability information. In addition topdating the system posterior reliability using the BMM , the subsystemosterior reliability distribution is also updated by integrating the sys-em level information through the proposed adaptive SIR algorithm.

.3.1. S1- system reliability estimation using system level data

When the system prior distribution and system level test data as wells the subsystem prior are available, the posterior distribution of systemeliability 𝜙 can be evaluated using the two methods as discussed inection 3.1.1 , i.e., integrating the pooled system prior distribution andhe updated system prior distribution with the likelihood function forhe system level data, respectively. In addition, the traditional Bayesiannference using the natural system prior with system data is comparedith these two Bayesian melding based methods. The pooled systemrior distribution 𝑞 ∗

𝜙( 𝜙) is derived from the induced system prior distri-

ution 𝑞 ∗ 1 ( 𝜙) and the natural system prior distribution q 2 ( 𝜙) as in Eq. (6) .he induced system prior distribution 𝑞 ∗ 1 ( 𝜙) is evaluated based on theubsystem prior distributions q 1 ( 𝜃) and the deterministic function M (·).he updated system prior distribution q 𝜙( 𝜙) is evaluation based on thepdated subsystem prior distributions q 𝜃( 𝜃) and the system structure,here q 𝜃( 𝜃) can be obtained as in Eq. (8) . To demonstrate the effectsf non-informative and informative priors on the posteriors in this case,he four above mentioned cases in terms of the availability of system andubsystem prior information are: 1) non-informative system and subsys-em priors; 2) non-informative system prior and informative subsystemrior; 3) informative system prior and non-informative subsystem prior;) informative system and subsystem priors. The effects of the sampleize of system data is also considered by using sample sizes of 5, 10,00, and 1000 for posterior system reliability inferences. Under a givenample size, 10,000 simulation runs are conducted through a binomialistribution with the success probability at 0.5 representing the reliabil-ty of 50%. For comparison purpose, the mode reliability and its 95%redible interval are used to characterize the posterior system reliability.

Case 1: non-informative system and subsystem priors

Assume both system and subsystem non-informative priors followeta (2, 2) distributions. Fig. 9 shows the posterior system mode relia-ility and 95% credible intervals, in which S 11 and S 12 refer to the firstnd the second method in Scenario 1 using pooled and updated systemrior, respectively.

From Fig. 9 , it can be seen that the mode reliability of system pos-erior reliability based on the updated system prior ( S 12) is closer tohe assumed reliability than that using the pooled system prior ( S 11) atll sample sizes. However, the 95% credible interval of the mode reli-bility based on the pooled system prior is narrower than that of the

153

eliability mode estimation using the updated system prior. This nar-ower credible interval can be explained as follows: the updated systemrior is derived using the system structure twice, i.e., for evaluating theooled system prior, and then for the updated system prior using the up-ated subsystem priors which is derived from the pooled system prior.uplicate prior information usage can result in larger posterior infer-nce uncertainty when only non-informative priors are available. Thus,he posterior mode reliability by integrating the non-informative priorshrough the updated system prior with available data has a wider cred-ble interval, while the mode of posterior using the pooled system prioras a narrower interval. The narrower credible interval indicates thathe estimation of the posterior distribution based on the pooled systemrior is more preferable when only non-informative priors exist. Theosterior based on the updated prior incorporate the system configura-ion twice, amplifying the effects of the subsystem prior. However, ashe sample size increases, both mode reliability and credible intervalsonverge to the same level.

Case 2: non-informative system prior and informative subsystem priors

The system non-informative prior is assumed to be Beta (2, 2) and in-ormative subsystem priors are given as Beta distributions which are pa-ameterized in terms of expert judgment for the mode reliability and itsonfidence level ( Table 2 ). In this case, it is observed that the posteriorystem reliability estimates using the pooled system prior and updatedystem prior are very close to each other, and are both greater than thessumed system reliability of 0.5. The 95% credible intervals using up-ated system prior is slightly wider than that of using the pooled systemrior. As the sample size increases, both the posterior mode reliabilitynd credible intervals converge to the same level ( Fig. 10 ).

Case 3: informative system prior distribution and non-informative sub-

ystem priors

In this case, the informative system prior is modeled as 𝐵𝑒𝑡𝑎 (0 . 96 ∗0 + 1 , 0 . 04 ∗ 10 + 1) based on the expert mode reliability and confidenceudgment ( Table 2 ) , and non-informative subsystem priors are as as-umed to be Beta (2, 2). From Fig. 11 , it is observed that the poste-ior system mode reliability estimations are close to each other and theode reliability using the traditional Bayesian method is higher than

he assumed system reliability. In addition, by comparing with the re-ults from Case 2, we observe that using informative system prior withigher mode system reliability is less influential to the system posterioreliability inference than using informative subsystem prior with higherode reliability.

Case 4: informative system and informative subsystem priors

The informative system and subsystem priors are given as Beta dis-ributions, which are parameterized using the expert reliability judg-ent for reliability model and confidence values shown in Table 2 .ompared with Cases 2 and 3, when both system and subsystem use

nformative prior information, the posterior system mode reliability es-imations are much higher than the assumed system reliability of 0.5,ut are closer compared with the mode reliability estimation using theraditional Bayesian method, see Fig. 12 .

For all four cases in S 1, the system posterior mode reliability usingpdated system prior ( S 12) is closer to the assumed system reliabilityhan that of using pooled system prior ( S 11). However, the 95% credi-le intervals using updated system prior are wider than those using the


Fig. 9. Mode reliability and the width of the 95% credible interval over various sample sizes.




154




p

b

s

v

t

i

4

s

t

d

s

u

E

t

d

b

s

u

l

c

t

f

u

p

b

T

i

n

a

d

r

a

n

a

r

s

a

f

g

p

c

B

i

i

s

i

B

e

i

F

v

m

a

c

o

a

o

ooled system prior for system posterior reliability inference. This cane attributed to the prior updating mechanisms of multiple usage of theubsystem prior information, See Eq. (2) , which may introduce higherariation in the posterior inference. It is also observed that the informa-ive subsystem priors are more influential to posterior system reliabilitynference than the effects from informative system prior information.

.3.2. S 2 and S 3- system and subsystem reliability estimation using

ubsystem level data only or both subsystem and system level data

In the cases of the existence of subsystem level data only, i.e., S 2,he subsystem posterior distribution can be evaluated based on the up-ated subsystem prior and the subsystem level data. First, the updatedubsystem prior distribution q 𝜃( 𝜃) is evaluated as in Eq. (3) . Then, thepdated subsystem posterior distribution 𝜋𝑠𝑢𝑏

𝜃can be estimated as in

q. (6) using the updated subsystem prior distribution and the subsys-em level data. The system posterior distribution 𝜋𝜙( 𝜙) can be furthererived from 𝜋𝑠𝑢𝑏

𝜃( 𝜃) based on the system structure. In the cases when

oth system level and subsystem level test data are available, i.e., S 3, theubsystem posterior distribution 𝜋𝑎𝑙𝑙

𝜃( 𝜃) can be evaluated based on the

pdated subsystem prior distribution and both system and subsystemevel test data as in Eq. (16) . The system posterior distribution 𝜋𝜙( 𝜙)an be derived from 𝜋𝑎𝑙𝑙

𝜃( 𝜃) based the system structure.

In S 2 and S 3, the posterior inference need to evaluate multiple quan-ities such as the induced prior, pooled prior, and the complex likelihoodunctions as seen in Eqs. (15) and (16) , and the adaptive SIR method issed. Similar to Scenario 1 for system posterior inference, the subsystemosteriors under four prior information combinations are investigatedy comparing the posterior mode reliability and its credible interval.he analysis uses 10,000 simulation runs, and the subsystem reliability

s assumed to be 0.5. When implementing adaptive SIR algorithm, theumber of original sampled importance weights is set to be 100,000,nd the number of selected weights and the percentage parameter for

155

etermining the threshold weight value are assigned as 1000 and 90%,espectively. The weight splitting process stops when the standard devi-tion of importance weights is smaller than 100. Similarly, four combi-ations of informative and non-informative system and subsystem priorsre considered to illustrate the effects of misspecified priors on poste-iors. To demonstrate the effect of the system level data on the sub-ystem posterior, the subsystem posterior inferences in Scenario 2 ( S 2)nd Scenario 3 ( S 3) are compared. Under the similar simulation setupor both system and subsystem level data, i.e., the pass/fail data areenerated from a binomial distribution with the reliability of 0.5. Theosterior mode reliability estimations under various sample sizes is alsoompared.

Case 1: non-informative system and subsystem priors

Assume both system and subsystem non-informative priors followeta (2, 2) distribution. Fig. 13 shows the posterior mode reliability and

ts 95% credible interval of subsystem 1 under different sample sizes. Its observed that the posterior mode reliability in S 3 is closer to the as-umed subsystem reliability than that of S 2. In addition, the 95% cred-ble intervals in S 3 is the smallest among S 2, S 3, and the traditionalayesian inference. With sample sizes increase the posterior reliabilitystimations converge.

Case 2: non-informative system and informative subsystem priors

The system non-informative prior is assumed to be Beta (2, 2) andnformative subsystem priors are given as Beta distributions ( Table 2 ).ig. 14 shows the posterior mode reliability and its 95% credible inter-al of subsystem 1 estimated under various sample sizes. The posteriorode reliability in S 3 is closest to the assumed subsystem reliability

mong S 2, S 3, and the traditional Bayesian inference method. The 95%redible interval from S 3 is also the narrowest among the three meth-ds except in extremely small size, i.e. only 5 pass/fail data points arevailable for both the system and subsystem. The posterior estimationsf subsystems 2, 4a, 4b, and 5 are similar with that of subsystem 1.




Fig. 17. Posterior modes from the pooled system prior with various pooling parameters.

1

j

s

i

r

S

u

p

S

T

t

p

t

i

t

S

B

e

e

t

Case 3: Informative system and non-informative subsystem priors

In this case, the informative system prior is modeled as 𝐵𝑒𝑡𝑎 (0 . 96 ∗0 + 1 , 0 . 04 ∗ 10 + 1) based on the expert mode reliability and confidenceudgment ( Table 2 ), and (non-informative subsystem priors are as as-umed to be Beta (2, 2). Fig. 15 shows the posterior mode reliability andts 95% credible interval of subsystem 1. Given informative system prioreliability information, the posterior mode reliability from both S 2 and 3 are shifted to higher reliability estimation, but the mode reliabilitysing the updated subsystem prior is less affected than that of using theooled subsystem prior in S 2. In addition, the 95% credible interval in3 is the narrowest among S 2, S 3, and the traditional Bayesian methods.he posterior estimation of subsystem 2, 4a, 4b, and 5 is similar with

m

156

hat of subsystem 1. Case 4: informative system and informative subsystem

riors

The informative system and subsystem priors are given as Beta dis-ributions ( Table 2 ). Fig. 16 shows the posterior mode reliability andts 95% credible interval of subsystem 1. Given both informative sys-em and subsystem priors, the posterior subsystem mode reliability in 3 is least affected among the three methods, S 2, S 3, and the traditionalayesian methods. The 95% credible interval of S 3 is also the narrowestxcept when extremely small sample size is used, i.e., 5. The posteriorstimation of subsystems 2, 4a, 4b, and 5 is similar with that of subsys-em 1.

For all four cases in S 2, it is observed that the subsystem posteriorode reliability is shifted up more when informative subsystem prior


i

a

t

n

a

b

i

f

3

i

t

d

4

u

i

b

i

o

r

o

t

t

o

o

w

F

5

(

m

d

t

b

s

s

o

c

m

H

P

s

t

a

W

t

r

i

a

o

c

m

s

s

i

m

p

p

m

t

s

t

a

w

p

t

a

t

f

d

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

nformation is used than when informative system prior information isvailable. Such observation is similar to the situations in S 1. In addition,he 95% credible intervals of mode reliability estimated by the BMM arearrower than that estimated by the traditional Bayesian method. Andll four cases in S 3, similar effects of posterior subsystem mode relia-ility estimation are noticed, e.g., posterior mode reliability estimations more sensitive to informative subsystem prior information than in-ormative system prior information (Cases 2 and 4 versus Cases 1 and). But the posterior estimation using updated subsystem prior ( S 3) us-ng both system and subsystem level data are the least affected amonghe S 2, S 3, and the traditional Bayesian method. The effects of usingifferent prior information diminish as the sample size increases.

.3.3. Effects of various pooling parameters on the posterior inference

sing BMM

The pooling parameter 𝛼 in Eq. (8) is essentially arbitrary within thenterval [0, 1] [30,31] .

One could determine the “optimal ” value of the pooling parameterased on optimality criteria, such as maximizing entropy and minimiz-ng Kullback–Liebler divergence in [32–34] . To understand the effectsf the various pooling parameters on the posteriors using BMM , poste-ior mode reliability and its 95% credible interval under the sample sizef 100 with informative system and subsystem priors are studied. Withhe informative priors and simulated data from two assumed true sys-em reliability of 0.50 and 0.96, it is observed that increasing the valuef the pooling parameter increases the mode of posterior distributionf the system reliability. Such increases can be explained as the highereighting of induced system reliability prior of 𝑞 ∗ 1 ( 𝜙) in this case, seeig. 17 .

. Conclusion

This paper investigates the applications of Bayesian Melding method BMM ) for system and subsystem reliability assessment/updating whenultilevel reliability information is available. The proposed BMM up-ates the subsystem prior reliability distribution by integrating the sys-em and subsystem prior distributions, and evaluates the posterior distri-ution with the updated subsystem prior distribution and system and/orubsystem level test data. The proposed BMM of updating the system andubsystem posterior reliability by integrating multilevel prior and vari-us data availability is investigated the first time in the literature. To ac-ommodate the computational challenges for posterior inference whenultiple priors and likelihood functions are involved, the Metropolis-astings algorithm and an adaptive SIR simulation algorithm are used.articularly, three scenarios in terms of the availability of system and/orubsystem data are discussed, and simulation is used to update both sys-em and subsystem level posterior reliability in terms of mode reliabilitynd 95% credible intervals.

From this study, the following four major findings are observed: 1)hen only system level data are available ( S 1), system posterior estima-

ions using updated system prior ( S 12) is closer to the assumed systemeliability with wider credible intervals than the reliability estimates us-ng the pooled system prior ( S 11). 2) When only subsystem level datare available ( S 2) and system priors are used to update subsystem pri-rs through the BMM , the subsystem posterior estimation has narrowerredible intervals than that estimated based on the traditional Bayesianethod. 3) When both system and subsystem data are available ( S 3), the

ubsystem posterior reliability estimates are closer to the assumed sub-ystem reliability compared with Scenario 2. In addition, the crediblentervals are the narrowest among S 2, S 3, and the traditional Bayesianethod without using using the BMM . 4) For both system and subsystemosterior reliability updating, it is observed that informative subsystemrior is more influential to posterior reliability inference than the infor-ative system prior information. However, the impacts of prior informa-

ion on the posterior inference diminishes as the sample size increases.

157

In this research, the deterministic function representing the systemtructure is illustrated through a mixed series-parallel structure. Whenhe system structures become more complex, the deterministic functions well as conditional functions will be more difficult to derive and thereill be no closed form for the inverse function of M (·). Under these com-lex and potential dynamic system structure, the authors speculate thathe differences of posterior inferences will be larger and more complexnd that will be a part of the future work. In addition, the current inves-igation uses the binary pass/fail data in the illustrative examples, anduture work will be extended to multiple types of data such as lifetimeata and degradation data.

eferences

[1] Meeker W , Escobar L . Statistical methods for reliability data. John Wiley & Sons;2014 .

[2] Hamada M , Wilson A , Reese CS , Martz H . Bayesian reliability. New York: SpringerScience+Business Media; 2008 .

[3] Martz H , Waller R , Fickas E . Bayesian reliability analysis of series system of bi-nomial subsystem and components. Am Stat Assoc Am Soc Qual Control Technom1988;30:143–54 .

[4] Martz H , Waller R . Bayesian reliability analysis of complex series/ parallel systemsof binomial subsystems and components. Technometrics 1990;32:407–16 .

[5] Burgman M , McBride M , Ashton R , Speirs-Bridge A , Flander L , Wintle B , et al. Expertstatus and performance. PloS ONE 2011;6(7):1–7 . E22998.

[6] O ’Hgan A , Buck C , Daneshkhah A , Eiser J , Garthwaite P , Jenkinson D , et al. Uncer-tain judgements: eliciting experts probabilities. John Wiley & Sons; 2006 .

[7] McConway K . The combination of experts opinions in probability assessment: sometheoretical considerations. University College London; 1978. Phd Thesis .

[8] Cooke R , Louis G . TU delft expert judgment data base. Reliability Engineering &System Safety 2008;93:657–74 .

[9] Madansky A . Externally bayesian Groups. University of Chicago; 1978 . Unpublishedmanuscript.

10] Poole D , Raftery A . Inference for deterministic simulation models: the bayesian meld-ing approach. J Am Stat Assoc 2000;95(452):1244–55 .

11] Winkler R . The consensus of subjective probability distribution. Manage Sci1968;15:B61–75 .

12] Hamada M , Martz H , Reese CS , Graves T , Johnson V , Wilson A . A fully bayesianapproach for combining multilevel failure information in fault tree quantificationand optimal follow-on resource allocation. Reliab Eng Syst Safety 2004;86:297–305 .

13] Reese CS , Wilson A , Guo J , Hamada M , Johnson V . A bayesian model for integratingmultiple sources of lifetime information in system-reliability assessments. J QualTechnol 2011;43(2):127–41 .

14] Peng W , Huang H , Xie M , Yang Y , Liu Y . A bayesian approach for system reliabilityanalysis with multilevel pass-fail, life time and degradation data sets. IEEE TransReliab 2013;62:689–99 .

15] Yontay P , Pan R . A computational bayesian approach to dependency assessment insystem reliability. Reliab Eng Syst Safety 2016;152:104–14 .

16] Guo J , Wilson AG . Bayesian methods for estimating system reliability using hetero-geneous multilevel information. Technometrics 2013;54(4):461–72 .

17] Wilson A , Graves T , Hamada M , Reese CS . Advances in data combination, analysisand collection for system reliability assessment. Stat Sci 2006;21(4):514–31 .

18] Keeney R , von Winterfeldt D . Eliciting probabilities from experts in complex techni-cal problems. IEEE Trans Eng Manage 1991;38:191–201 .

19] Wilson A , McNamar L , Wilson G . Information integration for complex systems. Re-liab Eng Syst Safety 2007;92:121–30 .

20] Rohatgi K . An introduction to probability theory mathematical statistics. New York:Wiley; 1976 .

21] Raftery A , Givens G , Zeh J . Inference from a deterministic population dynamicsmodel for bowhead whales. J Am Stat Assoc 1995;90:402–30 .

22] Wolpert R . Comment on “inference from a deterministic population dynamics modelfor bowhead whales. J Am Stat Assoc 1995;90:426–7 .

23] Yuan T , Ji Y . A hierarchical bayesian degradation model for heterogeneous data.IEEE Trans Reliab 2015;64:63–70 .

24] Rubin D . Bayesianly justifiable and relevant frequency calculations for the appliedstatistician. Ann Stat 1984;12:1151–72 .

25] Taddy M , Lee H , Sanso B . Fast bayesian inference for computer simulation inverseproblem. Inverse Probl 2009:8 .

26] Frigessi A , Martinelli F . Computational complexity of markov chain monte carlomethods for finite markov random fields. Biometrika 1997;87:1–18 .

27] Liang F . Dynamically weighted importance sampling in monte carlo computation. JAm Stat Assoc 2002;97:807–21 .

28] Yuan C , Druzdzel M . Improving importance sampling by adaptive split-rejection con-trol in bayesian network, advances in artificial intelligence. In: 20th Conference ofthe Canadian society for computational studies of intelligence, Canadian AI 2007,Montreal, Canada, May 28–30, 2007. Proceedings; 2007. p. 332–43 .

29] Li Z , Mense A . Bayesian reliability modeling for pass/fail systems with sparse data.The 61st annual reliability & maintainability symposium, Palm Harbor, FL, USA;2015 .

30] French S . Group consensus probability distributions: a critical survey. Bayesianstatistics, 183–201. Berger J, Bernardo J, Dawid A, Smith A, editors. North-Holland,Amsterdam: Oxford University Press; 1985 .

http://refhub.elsevier.com/S0951-8320(16)30388-X/sbref0001

















































































































[

[

[

[

31] Li Z , Guo J , Xiao N . Multiple priors integration for reliability estimation using thebayesian melding method. 63th annual reliability and maintainability symposium,Orlando, FL, USA; 2017 .

32] Savchuk V , Martz H . Bayes reliability estimation using multiple sources of priorinformation: binomial sampling. In: IEEE transactions on reliability, vol. 43; 1994.p. 138–44 .

158

33] Rufo M , Pérez C . Log-linear pool to combine prior distributions: a suggestion for acalibration-based approach. Bayesian Anal 2012;7(2):411–38 .

34] de Carvalho L., Villela D., Coelho F., Bastos L. Choosing the weights for the logarith-mic pooling of probability distributions. World Stat Congr 2015. ArXiv:1502.04206 .











http://arxiv.org/abs/1701.01352

reliability engineering and - university of...

Documents