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NSERC-UNENE Industrial Research Chair
Department of Civil and Environmental Engineering
University of Waterloo
Mahesh Pandey
Section 17
Bayesian Reliability
Analysis
Learning Objectives
The purpose of this section is to:
⚫ Demonstrate how to perform reliability computations when little or
no data is available
⚫ Discuss the idea behind “Weibayes” analysis, and why it is not
really a Bayesian approach
⚫ Introduce the concept of material improvement factors (MIFs)
used in the reliability assessment of SG tubing materials
⚫ Show how to estimate and update important reliability parameters,
such as the constant failure rate l, using Bayesian “conjugate”
distributions
⚫ Describe how industry generic data can readily be used to
construct conjugate “prior” distributions
⚫ Explain the key challenges and advantages associated with the
Log-normal prior distribution used commonly in PSA
2
Introduction
3
Introduction
Often the most challenging part of a credible reliability analysis
is the lack of data
⚫ Most engineering systems, structures and components (SSCs)
are highly reliable by design
⚫ Preventive maintenance (PM) programs are also in place to
mitigate against failures
⚫ Reliability testing of new components may be expensive both in
terms of cost and duration
The objective of Bayesian analysis is to supplement the lack
of data with expert judgment and/or generic information
There are two basic types of reliability problems
⚫ Analysis of first failures
⚫ Analysis of repeat failures
4
First Failure Analysis (Weibull)
The analysis of first failure is typically based on the Weibull
distribution
Basic idea is to fit the Weibull distribution to the time to (first)
failure data
⚫ Including censoring (i.e., survivors)
If little or no data is available, assumptions must be made
regarding the Weibull shape and scale parameters
⚫ “Weibayes”: Assume the shape parameter is known
⚫ Fully Bayesian: Assign a prior distribution to both parameters
This approach is most applicable to the survival or reliability
analysis of groups of similar (i.e., identical) components
⚫ Used in product design and manufacturing
5
Repeat Failure Analysis (HPP)
The analysis of repeat failures (or the frequency of recurring
events) is typically based on the Poisson process (HPP)
model
Here the interest is in analyzing the number of failures in a
given time interval
⚫ Failures are assumed to be fixed to as-good-as-new condition in a
short time interval
If little or no data is available, assumptions must be made
regarding the parameter l in the Poisson distribution
⚫ Fully Bayesian analysis using a conjugate (Gamma) prior
This approach is most applicable to the analysis of systems in
continuous operation (i.e., repairs are being made)
⚫ Basis for Probabilistic Safety Assessment (PSA)
6
PSA Analysis
The Bayesian approach is very useful in PSA
⚫ Analysis of low probability events for which few data are available
Both expert judgment and generic data are used for
characterizing prior distributions
Site specific information is then used to update the prior
distributions to obtain the posteriors
Generic data typically allows the estimation of prior distribution
parameters
⚫ Available from many industry sources and databases (e.g., INPO,
EPRI, NRC, DOE, etc.)
⚫ Allows the use of Bayesian conjugates
Expert judgment in PRA is often incorporated using the
Log-normal distribution with an error factor
7
Bayesian Update of Failure
Probability
88
Failure Probability (or Proportion)
The parameter p in the Binomial distribution represents the
probability of failure or proportion
⚫ From data, p can be estimated simply by dividing the observed
number of failures x with the observed number of trials n
⚫ n can also represent the number of identical components in a fleet
(i.e., population size)
If data is scarce or not available, we can treat p as a random
variable and assign a probability distribution to it
It is possible to use any type of prior distribution for p, however,
it is computationally advantageous and preferable to use a
conjugate distribution
The conjugate distribution for the Binomial distribution is the
Beta distribution
9
Beta/Binomial Conjugate
Therefore, we describe the parameter p as a random variable
having the Beta distribution
This is the prior distribution for the parameter p
⚫ i.e., probability of failure
The mean and variance of p are equal to
The unknown distribution parameters are a and b where
⚫ a corresponds to the prior number of failures
⚫ b corresponds to the prior number of successes (or no failures)
(no Excel formula available for the PDF)
10
Beta/Binomial Conjugate (cont’d)
The likelihood function in this case is the Binomial distribution
The evidence (i.e., observed data) are x and n where
⚫ x is the number of failures
⚫ n is the number of trials (or components in the fleet)
The updating is based on the Bayes formula
However, because we are using conjugate distributions, there
is no need to evaluate the Bayes’ theorem explicitly
⚫ i.e., no integration is required
=BINOMDIST(x,n,p,FALSE)
11
Beta/Binomial Conjugate (cont’d)
The posterior distribution is also a Beta distribution
The parameters of the posterior distribution are obtained using
simple formulas as
Therefore, the parameters of the prior distribution can be
updated directly using the new information
⚫ i.e. there is no need to use the integral form of the Bayes’ theorem
x is the number of observed failuresn is the number of trials (or components)
12
Beta/Binomial Conjugate (cont’d)
The credibility interval for p can then be obtained by
computing the appropriate quantiles from the posterior Beta
distribution
⚫ For example, for the
90 % credibility interval
The median value of p can
also be obtained as
Median Upper
Limit
Lower
Limit
Area = 0.05Area = 0.05
13
Consider the failure to start of the turbine train of the auxiliary
feedwater system (discussed in a previous example). Nine years
of industry data from such an event has been analyzed and
compiled in NUREG/CR-5500 Vol. 1. Based on the analysis, it
was determined that on average, there were 4.2 failures of the
turbine train to start in 157.3 demands.
(a) Plot the prior distribution of the failure to start assuming it
follows the Beta distribution.
(b) Compute the posterior distribution for a plant where the train
has failed to start once in the last eight demands.
(c) Compare the results of the Bayesian approach to the
maximum likelihood estimate using the plant data only.
Example 1
Solution:
⚫ Part (a) The prior parameters of the Beta distribution are a - the
number of failures and b - the number of successes (no failures)
14
Solution (cont’d)
For the generic industry data, we have x = 4.2 (number of failures) and n = 157.3 (number of demands)
The number of successes (no failures in 157.3 demands) is therefore equal to
Therefore, for the generic industry data, we get a = 4.2 and b = 153.1
The prior distribution for the failure to start p is given by the Beta distribution
where a = 4.2 and b = 153.1
15
Solution (cont’d)
The prior distribution is plotted below
0
5
10
15
20
25
30
35
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
p (failures/demand)
De
ns
ity
Mean
= 0.0267
Median
= 0.0247
a = 4.2
b = 153.1
16
Solution (cont’d)
Part (b) We now have to consider the observations from the
actual plant.
⚫ We have x = 1 (number of failures) in n = 8 (number of demands)
⚫ Because of the Beta/Binomial conjugate, the parameters of the
posterior distribution for p are easy to compute as
The posterior distribution for p is therefore given by
where a´ = 5.2 and b´ = 160.1
17
Solution (cont’d)
The mean and the median of the posterior distribution are
equal to
The 90 % credibility interval for the posterior distribution is computed using the =BETAINV() function as
18
0
5
10
15
20
25
30
35
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
p (failures/demand)
De
ns
ity
Prior Distribution
Posterior Distribution
Max. Likelihood (data only)
Solution (cont’d)
Part (c) Compare the results to the plant data only.
⚫ The max. likelihood estimate is equal to p = 1/8 = 0.125
⚫ The 90 % confidence interval was computed previously in
Example 15.1 and is equal to 0.0064 < p < 0.47
Very strong prior!!!
(very few data)
19
Solution (cont’d)
The results are summarized below
Distribution 5th %tile Mean Median 95th %tile
Prior 0.00954 0.0267 0.0247 0.0506
Posterior 0.0128 0.0315 0.0296 0.0565
Plant Data Only 0.0064 0.125 -- 0.47
0.1250
0.0315
0.0267
0 0.1 0.2 0.3 0.4 0.5
p (failures/demand)
Prior
(industry average)
Posterior
(Bayesian estimate)
Max. LikelihoodPlant Data
Only
20
Bayesian Update of Failure
Rate
21
Poisson Process
The occurrence of recurring events (e.g., repeat failures) is
often modelled as a Homogeneous Poisson Process (HPP)
⚫ The failure rate l is constant (i.e., independent of time)
⚫ Repair time is assumed to be negligible
The probability associated with the number of failures in a
given time interval is given by the Poisson distribution
⚫ It is also the likelihood function for the Bayesian approach as
The parameter of interest is the unknown l which represents
the (constant) occurrence (e.g., failure) rate
⚫ i.e., “average” number of failures per unit time
=POISSON(x,lt,FALSE)
22
Gamma/Poisson Conjugate
The Bayesian conjugate distribution for the Poisson
distribution is the Gamma distribution
Therefore, we describe the constant failure rate l as a random
variable having the Gamma distribution
The probability density function (PDF) for the Gamma
distribution is
(Note that the Excel function uses 1/b instead of b)
This is the prior distribution for the failure rate l
The unknown distribution parameters are a and b where
⚫ a corresponds to the total number of failures
⚫ b corresponds to the total operating time
=GAMMADIST(l,a,1/b,FALSE)
23
Gamma/Poisson Conjugate (cont’d)
Because the Gamma distribution and Poisson distribution are
conjugates, the posterior distribution is also a Gamma
distribution
The mean and variance of l are given as
The parameters of the posterior distribution are obtained using
simple formulas as
Similar to before, the parameters of the prior distribution can be
updated directly using the new information
x is the number of observed failurest is the total operating time
=GAMMADIST(l,a’,1/b’,FALSE)
24
Gamma/Poisson Conjugate (cont’d)
The credibility interval for l is obtained by computing the
appropriate quantiles from the posterior Gamma distribution
⚫ For example, for the
90 % credibility interval
The median value of l can
also be obtained as
The mean value is simply Median Upper
Limit
Lower
Limit
Area = 0.05Area = 0.05
=GAMMAINV(0.05,a′,1/b′)
=GAMMAINV(0.95,a′,1/b′)
=GAMMAINV(0.50,a′,1/b′)
25
Consider again the plant planning on upgrading their eddy
current inspection system. Rather than a pure expert judgment,
the system engineer decides to use the failure history of the old
equipment to construct the prior distribution for the new
technology. Based on the historical data, the engineer has
estimated that the old system failed on average 3.2 times per
each outage campaign.
a) Construct and plot the prior distribution for the equipment
failure rate, assuming it follows the Gamma distribution.
b) Compute the posterior distribution for the failure rate given
that 2 failures of the new system were observed in the most
recent plant outage.
Example 2
Solution:
⚫ Assume the failure of the eddy current inspection system follows
the Homogeneous Poisson process (HPP)
26
Solution (cont’d)
Part (a): The prior parameters of the Gamma distribution are
⚫ a - the number of events and
⚫ b - the operating time
Using the historical data, we have a = 3.2 (number of failures)
and b = 1 (outage duration)
The prior distribution describes the uncertainty in the Poisson
rate parameter l, which is the equipment failure rate
The prior distribution for l is therefore given by the Gamma
distribution
where a = 3.2 and b = 1
=GAMMADIST(l,a,1/b,FALSE)
27
The prior distribution is plotted below
Solution (cont’d)
28
Solution (cont’d)
Part (b): We now consider the observations from the latest
outage.
⚫ We have x = 2 failures in t = 1 outage
⚫ Because of the Poisson/Gamma conjugate, the parameters of the
posterior distribution for l are extremely easy to compute as
The posterior distribution for l is therefore given by
where a´ = 5.2 and b´ = 2
=GAMMADIST(l,a’,1/b’,FALSE)
29
Solution (cont’d)
The mean and the median of the posterior distribution are
equal to
The 90 % credibility interval for the posterior distribution is computed using the =GAMMAINV() function as
30
The two distributions are plotted below for comparison
⚫ The plant data only estimate is equal to l = 2/1 = 2 failures/outage
⚫ The posterior distribution has less uncertainty (i.e., spread) and
has moved toward the point estimate
Solution (cont’d)
31
Reliability Analysis
Based on the previous example, what is the probability of
having no failures in the next outage?
We know that the probability is computed using the Poisson
distribution, however, now the failure rate l is no longer
constant but follows the Gamma distribution
⚫ This means that the probability of no failure in the next outage is
no longer a single value but a distribution!
May be difficult to evaluate analytically
Use the “best estimate” approach
⚫ e.g., use the mean or median failure rate as the “best estimate”
⚫ Could also use some upper percentile for worst case risk etc.
(i.e., bounding analysis)
32
Log-Normal Prior
33
Lack of Evidence
It is evident that there are many benefits for selecting prior
distributions that are conjugate to the failure models
⚫ Bayesian updating is straightforward
⚫ The distributions have tidy algebraic formulas
⚫ The information is easily entered into a PRA program by simply
entering the distribution type and associated parameter values
In some cases, however, there may not be enough evidence
(e.g., observations of failure) to estimate the prior distribution
type and its parameters
Must use expert opinion and engineering judgment
The Log-Normal distribution is often used to in this case
⚫ The Log-Normal prior is used extensively in PRA
34
Log-Normal in PRA
Historically, basic event inputs in Probabilistic Risk Assessment
(PRA) have been characterized using the Log-Normal
distribution (e.g., WASH-1400)
The use of the Log-Normal distribution is attractive because
⚫ It is always positive (good for modelling non-negative phenomena)
⚫ Has two parameters (allows for flexibility)
⚫ Large amount of existing failure rate knowledge, both plant-
specific and generic industry data is found in this form
Unfortunately, the Log-Normal distribution is not a conjugate to
any of the other standard distributions
⚫ Makes Bayesian analysis more complicated
⚫ i.e., no simple update formulas exist
35
Error Factor
The uncertainty range in the expert estimate is typically
characterized using the error factor
The error factor for the Log-Normal distribution is defined
simply as the ratio of the 95th percentile value to the median
(or the ratio of the median value to the 5th percentile value)
The error factor describes the amount of “dispersion” of spread
in the distribution from the median value
⚫ For example, for EF = 10, the lower and upper bounds of the
90 % confidence interval are equal to “10 times” the median value
⚫ E.g., for l = 10-3, the 90 % confidence interval is 10-4 < l < 10-2
36
0
100
200
300
400
500
600
700
800
0 0.001 0.002 0.003 0.004 0.005l
De
ns
ity
l 50 = 0.001
l 95 = 0.003l 05 = 3.3E-4
EF = 3
Error Factor (cont’d)
Log-Normal distribution for l = 10-3 and EF = 3
37
Error Factor (cont’d)
In most cases, the Log-Normal prior is simply characterized by
the median value and an error factor
⚫ e.g., an initiating event frequency may have a point estimate
equal to 10-4 (assumed to be the median value) with an error
factor of 5
Error factors in PRA generally range from 1.3 to 30
⚫ Smallest factors or 1.3 - 2 typically apply to the higher event
frequencies (one or more per reactor year) for which a larger
amount of recorded data generally exists
⚫ Factors of 2 - 3 apply to higher valued component failure rates
and to single human error rates, which are in the vicinity of 1×10-3
per demand or per attempt
⚫ The largest factors of 20 - 30 apply to unlikely pipe rupture rates
and multiple human errors being committed
38
