Bayesian Statistics Applied to Reliability Part

1 Rev. 1

- Allan Mense, Ph.D., PE, CREPrincipal Engineering

Fellow - Raytheon Missile Systems
- Tucson, AZ

What is Bayesian Statistics?

- It is the application of a particular probability

rule, or theorem, for understanding the

variability of random variables i.e. statistics. - The theorem of interest is called Bayes Theorem

and will be discussed in detail in this

presentation. - Bayes Theorem has applicability to all

statistical analyses not just reliability.

Blame It All on This Man

References.The two major texts in this area are

Bayesian Reliability Analysis, by Martz

Waller 2 and more recently Bayesian

Reliability, by Hamada, Wilson, Reese and Martz

1. It is worth noting that much of this early

work was done at Los Alamos National Lab on

missile reliability and all the above authors

work or have worked at the LANL 1. There are

also chapters in traditional reliability texts

e.g. Statistical Methods for Reliability Data,

Chapter 14, by Meeker and Escobar

Rev. Sir Thomas Bayes (born London 1701, died

1761) had his works that include the Theorem

named after him read into the British Royal

Society proceedings (posthumously) by a colleague

in 1763.

Bayesian Helps

Bayesian methods have only recently caught on due

to the increased computational speed of modern

computers. It was seldom taught to engineers

because there were few ways to complete

calculations for any realistic systems.

Background Probability

- To perform statistics one must understand some

basic probability - Multiplication rule Two events A,B the

probability of both events occurring is given by - P(A and B)P(AB)P(B) P(B and

A)P(BA)P(A)P(A?B) - There is no implication of time series of these

events - PAB) is called a conditional probability
- P(A and B) P(A)P(B) if A and B are independent,

- P(A and B)0 if A and B are mutually exclusive
- Example Have 3 subsystems in series the

probability the systems does not fail by time T

is RSYSR1(T)R2(T)R3(T) if failures are not

correlated

These rules also work for probability

distributions

Background

- To perform statistics one must understand some

basic probability - Addition rule Two events A,B the probability of

either event A or B or both occurring is given by - P(A or B)P(A)P(B) P(A and B) P(A?B)
- There is no implication of time series of these

events - P(A or B)P(A)P(B) P(A)P(B) if A and B are

independent, - P(A or B)P(A)P(B) if A and B are mutually

exclusive - Example Have 2 subsystems in parallel the

probability the systems does not fail by time T

is RSYSR1(T)R2(T)-R1(T)R2(T) if failures are

not correlated

These rules also work for probability

distributions

Background

- To perform statistics one must understand some

basic probability - Bayes Rule Two events A,B the probability of

event A occurring given event B occurs is given

by - P(A B)P(BA) P(A) / P(B)
- There is no implication of time series of these

events. - P(AB) posterior probability.
- P(A) prior probability.
- P(B)P(BA)P(A)P(BA) P(A), called the

marginal probability of event B also called the

Rule of Total Probability.

These rules also work for probability

distributions

Problem 1

- Aevent that missile thruster 1 will not

work0.15 - Bevent that missile thruster 2 will not work

0.15 - P(A or B or both A and B) P(A)P(B)-P(A)P(B)

0.150.15-(0.15)(0.15)

0.30-0.0225 0.2775 - P(A or B but not both A and B)

P(A)P(B)-2P(A)P(B) 0.15 0.15 0.045

0.255

Problem 2

- Let A man age gt 50 has prostate cancer, B

PSA score gt5.0, Say on the demographical region

of interest P(A) 0.3 that is the male

population over 50 has a 30 chance of having

prostate cancer. - Now I take a blood test called a PSA test and

that test supposedly helps make a decision about

the presence of cancer in the prostate. A score

of 5.5 is considered in the medium to high zone

as an indicator (actually doctors look at rate of

increase of PSA over time). - The PSA test has the following probabilities

associated with the test result P(BA)0.9,

P(BA) 0.2 i.e. even if you do not have cancer

the test registers PSAgt5.0 about 20 of the time

(false positive), Thus the probability of getting

a PSA score gt 5.0 is given by P(B)P(BA)P(A)P(B

A)P(A) (0.9)(0.3)(0.2)(0.7)0.41 - Using Bayes Theorem P(AB) P(BA)P(A)/P(B)(0.9)

(0.3)/0.410.66 - Your probability of having prostate cancer has

gone from 30 with no knowledge of test to 66

given your knowledge that your PSA gt 5.0

Bayesian in a nutshell

- Bayesian models have two parts the likelihood

function and the prior distribution. - We construct the likelihood function from the

sampling distribution of the data, which

describes the probability of observing the data

before the experiment is performed, e.g. binomial

distribution P(sn,R). This sampling distribution

is called the observational model. After we

perform the experiment and observe the data, we

can consider the sampling distribution as a

function of the unknown parameter(s), e.g.

(Rn,s) Rs(1-R)n-s. This function is called the

likelihood function. The parameters in the

observational model (R) are themselves modeled by

what is called a structural model e.g.. - The prior distribution describes the uncertainty

about the parameters of the likelihood function

(R Nm, p) RNm p(1-R)Nm (1-p). The parameters

Nm and p are called hyperparameters (the

parameter in the original problem is R, the

reliability itself) - HIERARCHICAL MODELS Sometimes the parameters,

e.g. Nm and p, are not known and have their own

distributions known as hyperhyperdistributions or

hyperpriors which have their own parameters e.g.

Nm GAM( a, k). This is more complex but can lead

to better answers. - We update the prior distribution to the posterior

distribution after observing data. We use Bayes'

Theorem to perform the update, which shows that

the posterior distribution is computed (up to a

proportionality constant) by multiplying the

likelihood function by the prior distribution. - (Posterior of parameters) (Likelihood function)

X (prior of parameters)

Now we need to study each part of this equation

starting with the likelihood

Binomial Distribution for the Data

- Probability of an event occurring p
- Probability of event not occurring 1-p
- Probability that event occurs x2 times in n5

tests - 1st instantiation pp(1-p)(1-p)(1-p) p2(1-p)3

(multiplication rule) - 2nd instantiation p(1-p)p(1-p)(1-p) p2(1-p)3
- 10th instantiation (1-p)(1-p)(1-p)pp p2(1-p)3
- In general the number of ways of having 2

successes in 5 trials is given by 5!/(3!2!) 10

number of combinations of selecting two items

out of ten. - nCr n!/((n-r)!r!)

Binomial distribution

- Since any of the 10 instantiations would give the

same outcome (i.e. 2 successes in 5 trials each

with probability p2(1-p)(5-2)) we can determine

the total probability using the addition rule for

mutually exclusive events. - P(1st Instantiation) P(2nd instantiation)

P() P(10th instantiation) 5C2 p2(1-p)(5-2) - P(xn, p) nCx px(1-p)(n-x)
- The random variable is x, the number of

successful events

binomial distributionrandom variable is x, the

number of successes

Likelihood Function

- If we have pass/fail data the binomial

distribution is called the sampling distribution

however, after the experiments or trials are

performed and the outcomes are known we can look

upon this sampling distribution as a continuous

function of the variable p. There can be many

values of p that can lead to the same set of

outcomes (e.g. x events in n trials). The core of

the binomial distribution -- that part which

includes p -- is called a likelihood function. - L(px, n) likelihood that the probability of a

successful event is p given the data x and n. - L(px, n) px (1-p)(n-x)
- The random variable is now p with x and n known

The Likelihood function is put to great use in

Bayesian statistics

Likelihood Function, L(ps,n)The random variable

is p, the probability of a success

Sometimes we label p as R, the reliability

Likelihood Functions What they look like!

1 Test and 0 Successes

1 Test and 1 Success

Likelihood function normalized to area under

function 1. This is not necessary but allows

for a better interpretation

Likelihood Functions for n2 tests

2 successes

0 successes

1 success

Bayesian Approach

- Use Bayes Theorem with probability density

functions e.g. - P(AB) gt fposterior(parameterdata) P(BA) gt

Likelihood(data parameter) P(A) gt

fprior(parameter before take data) - fposterior ? Likelihood X fprior
- Steps
- 1. determine a prior distribution for parameters

of interest based on all relevant information

before (prior to) taking data from tests. - 2. perform tests and insert results into

appropriate Likelihood function. - 3. multiply prior distribution times Likelihood

function to find posterior distribution. - Remember we are determining the distribution

function of one or more population parameters of

interest. - In pass/fail tests the parameter of interest is

R, the reliability itself. - In time dependent (Weibull) reliability analysis

it is a and b, the scale and shape parameters

,for which we need to generate posterior

distributions.

Summary on Probability

- Learned 3 rules of probability. Multiplication,

addition and Bayes. - Derived the binomial distribution for probability

of x events in n (independent) trials. - Defined a Likelihood function for pass/fail data

by changing the random variable in the binomial

distribution to the probability p with known data

(s,n). - Illustrated Bayes Theorem with probability

density functions.

Reliability

- General Bayesian Approach
- Reliability is NOT a constant parameter. It has

uncertainty that must be taken into account. The

language of variability is statistics. - Types of reliability tests
- Pass/Fail tests
- Time dependent tests (exponential Weibull)
- Counting experiments (Poisson statistics)

Bayes Theorem for Distributionswhen using

pass/fail data

- fposterior(Rn, s, info)L(s, nR) X fprior(R

info) - Example For pass/fail data
- fprior(R) ? Ra-1(1-R)b-1 (beta)
- L(s,nR) ? Rs(1-R)n-s (binomial)
- fposterior(R s, n, a, b) ? Rsa-1(1-R)n-sb-1

(beta-binomial)

Much more will be said of this formulation later

Prior, Likelihood, PosteriorWhat they look like!

You want the information that comes from knowing

the green curve!

Bayesian Calculator

In the following charts I want to take you

through a series of What if cases using priors

of different strengths and show how the data

overpowers the prior and leads to the classical

result when the data is sufficiently

plentiful. You will see that, in general, the

Bayesian analysis keeps you from being over

optimistic or over pessimistic when the amount of

data is small.

Bayesian gives lower probabilities for high

success and higher values for low success.

Bayesian keeps one from being too pessimistic or

too optimistic

Formulas Used In Beta-binomial model for

pass/fail data

- Prior Distribution
- Likelihood
- Posterior

The prior uses what I call the Los Alamos

parameterization of the beta distribution. See

Appendix for more detail

p0.9, Nm10

Take tests n4 and resulting successes s3

Note how strong prior distribution (Nm 10 which

is large) governs the posterior distribution when

there are a small number of data points. The

posterior has a maximum value at R0.857 where a

classical calculation would give ltRgt3/40.75.

p0.9, Nm2

Take tests n4 and resulting successes s3

Note how weak prior distribution (Nm2) will

still pull the posterior up that its net effect

is much smaller i.e. the posterior has a maximum

value at R0.800 where a classical calculation

would give ltRgt3/40.75. So the posterior and

likelihood are more closely aligned.

p0.9, Nm10

Increase tests Take tests n20 and resulting

successes s15

Note how strong prior distribution still pulls

the posterior up but due to the larger amount of

data (n20) the posterior distribution still

shows a maximum value at R0.800 so the prior is

beginning to be overpowered by the data.

p0.9, Nm2

Take tests n20 and resulting successes s15

Note how weak prior distribution still has some

effect but due to the larger amount of data

(n20) the posterior distribution is almost

coincident with the likelihood (i.e. the data)

and shows a maximum value at R0.764 which is

close to the data modal value of 0.75. The data

is overpowering the prior.

Nm0 uniform prior

Take tests n20 and resulting successes s15

Note how uniform prior distribution has no effect

on the posterior which is now exactly coincident

with the likelihood function and is essentially

the result you obtain using classical analyses.

Posterior mean R0.750 . The data is all we have.

Note that p plays no role because all values of R

are equally likely when Nm0. This is NEVER a

useful prior to use because it brings no

information to the analysis and it is this

additional information that drove us to use

Bayesian analyses in the first place.

Look at Predictions i.e. successes in the next

10 tests

p0.9, Nm10, n20, s15, ltRgt15/200.75

The Bayesian calculation shows that due to the

strong prior (Nm10) the prediction of success is

higher for large numbers of successes as compared

to the use of the average reliability in the

binomial calculation

p0.9, Nm10, n4, s3, ltRgt3/40.75

Given these Bayesian results how does one choose

a prior distribution?

- This is the most often asked question both by new

practitioners and by customers. - The answer is actually it depends.
- Depends on how you wish to represent previous

knowledge - Depends on your estimation technique for

determining the hyperparameters (p,Nm) e.g. what

degree of confidence, Nm ,you have in the

estimation of p. - What are some possible methods for guessing a p

value and setting an Nm value? - Are there other shapes of priors that one could

use? YES - What if we have uncertainty in Nm?
- Set up a hyperprior distribution for Nm say a

gamma distribution. - The parameters of f(Nm) i.e. (h,d) are called

hyperhyperparameters. - How do you choose the parameters d and h for the

gamma distribution? Pick mode for Nm (d-1)/h

and stdev of Nm d1/2 / h. - Example for a weak prior we might choose the

mode 3 for Nm, and a stdev 2, which gives

h1, d4

Now it gets complicated!

- Up until now we have been able to solve for

fposterior(R) analytically but once we have a

reasonable prior for Nm we no longer can do this.

Now the joint prior for R and Nm given p, h, and

d is shown below - There is no analytical solution for fposterior(R)

since one now must integrate over Nm. - This is one of the reasons that Bayesian has not

been widely accepted in a community that is

looking for simple recipes to calculate

reliability.

Will apply MCMC to find fposterior(R), learn this

in part-2 of lectures

Summary

- We see how Bayesian analysis for pass/fail data

can be analyzed rather easily for specific forms

of prior distributions --- called conjugate

priors. - We see how data overcomes the prior distribution

and the amount of data depends on the strength

of the prior. - We have ended with the formulation of a more

complex Bayesian analysis problem that will

require some form of numerical technique for its

solution

Numerical Techniques will be the subject of Part

2 of these lectures

Time Dependent Reliability Analysis

- Consider the case of a constant failure rate

time-to-first-failure distribution i.e.

f(t)lexp(-lt) - In classical analysis we look up values for l for

all the parts then perform a weighted sum of the

l values over all the components in the system

(series system) to arrive at a total constant

failure rate, lT, for the system. We can then use

lT to find the reliability at a given time t

using R(t)exp(-lTt). - What if there is uncertainty in the l values that

go into finding lT? How do we handle that

variability?

Applying a prior distribution to the parameter l

is the Bayesian process

Time Dependent Reliability Analysis

- Another technique applies when we have

time-to-first-failure data i.e. say we have n

units under test and the test is scheduled to

last for time tR. When conducting these tests

say r units fail and we do not replace the failed

units. The failure times are designated by ti,

i1,2,,r and (n-r) units do not fail by time tR. - The likelihood function for this set of n tests

is given by L lexp(-lt1) lexp(-lt2)

lexp(-ltr) X exp(-ltr1) exp(-ltr2)

exp(-ltn) lr exp(-l(t1t2tr (n-r)tR))

lr exp(-l(TTT)), TTTtotal time on test. - In classical analysis we differentiate the above

equation w.r.t. the parameter l and set the

derivative 0 and solve for the l value that

maximizes L (or more easily maximize ln(L)). The

value so obtained, lMLE r/TTT, is called the

Maximum Likelihood Estimate for the population

parameter l.Note For this distribution ONLY the

estimator for the failure rate does not depent on

the number of units on test except through TTT. - Again this technique assumes there is a fixed but

unknown value for l and it is estimated using the

MLE method. - But in real life there is uncertainty in l so we

need to discuss some distribution of possible l

values i.e. find a prior and posterior

distribution for l and let the data tell us about

the variability. The variability in l shows up

as a variability in R since Rexp(-lt). So for

any given time say tt1 we will have a

distribution of R values and that will be the

same distribution but with lower mean value for

later times. If l itself changed with time then

we have added as yet another complication.

Applying a prior distribution to the parameter l

is the Bayesian process

Exponential distribution

- Start with an assumed form for the prior

distribution for l. - One possible choice that allows for l to vary

quite a bit is the gamma distribution. Gamma(a,b) - With reasonable choices for a and b this

distribution allows for l to range over a wide

range of values. - Likelihood is given by

Exponential time-to-failure DistributionModel

variation of l with Gamma distribution

- Multiplying Prior X Likelihood gives
- So the prior is a Gamma ( a, b) distribution and

the Likelihood is the product of exponential

reliabilities for n tests run for long enough to

get n failures and so the we know all n failure

times. The posterior turns out to also be a

Gamma(na, bTTT), TTT(t1t2tn)total time on

test. - The problem that may occur here is in the

evaluation of the Gamma posterior distribution

for large arguments. One may need to use MatLab

instead of excel.

R(t)

- To find the reliability as a function of time one

must integrate over l from 0 to 8, i.e. - Again this can be integrated only for special

values of the parameters but evaluation for large

arguments of the Gamma distribution may require

top of the line software.

This will be evaluated later.

Weibull distribution(See Hamada, et al. Chapter

4, section 4)

- Once we have dealt with the exponential

distribution then the next logical step is to

look at the Weibull distribution that has two

parameters (a, b) instead of the single parameter

(l) for the exponential distribution. Now lets

address a counting problem which is very typical

of logistics analysis. With two parameters we

will need a two variable or joint prior

distribution fprior(a, b) which in some cases can

be modeled by the product fa,prior(a)fb,prior(b)

if the parameters can be shown to be independent.

- Even if independence cannot be proven one almost

always uses the product to make the priors

useable for modeling purposes.

Summary of time dependent Bayesian Reliability

Modeling

- It has been shown how to set of a typical

time-to-first-failure model. - The principals are the same as for any Bayesian

reliability model and there will be a

distribution of probability of failure vs time

for each time. - These distributions in parameters lead to

distributions in reliabilities and in principal

one can graph say 90 credible intervals for each

time of interest and these intervals should be

smaller than classical intervals that use

approximate normal or even nonparametric bounds.

Discussion of Hierarchal Models

- This follows closely the paper by Allyson Wilson

3 - Hierarchical models are one of the central tools

of Bayesian analysis. - Denote the sampling distribution as f(yq) e.g.

f(sR) binomial distribution, and the prior

distribution as g(qa) where a represents the

parameters of the prior distribution (often

called hyperparameters) e.g. g(RNm,p) beta

distribution. - It may be the case that we know g(qa)

completely, including a specific value for a.

However, suppose that we do not know a, and that

we choose to quantify our uncertainty about a

using a distribution h(a) (often called the

hyperprior), e.g. h(Nm) since we know p but do

not know Nm.

General Form for Bayesian Problem

- 1. The observational model for the data.
- (Yi q) ? f(yi q) i 1, , k
- 2. The structural model for the parameters of the

likelihood. - (qa) ? g(qa)
- 3. The hyperparameter model for the parameters of

the structural model. - a ? h(a)

Observational Model

- In the pass/fail example used early in this

presentation the observational model was the

binomial distribution f(s R) Rs (1-R)n-s where

the random variable is the number of successes,

s. - In this particular model the parameter of

interest, R, also happens to be the result we are

seeking. In other problems e.g. time to failure

problem using a Weibull distribution as the

observational model we have a and b as parameters

and we construct R(ta,b) exp(-(t/a)b), which

is the result we are seeking.

Structural Model

- The structural model addresses the variability of

the parameter q to a set of hyperparameters a. - In our pass/fail example we have a beta

distribution for our structural model or prior,

e.g. f(Rp,Nm) RNmp(1-R)Nm(1-p) - This parameterization of the beta distribution is

not the standard form which is typically written

as Ra-1(1-R)b-1 but I have chosen to use the Los

Alamos parameterization for its convenience of

interpretation (a Nmp1, b Nm(1-p)1),

because p mode of the distribution and Nm is a

weighting factor indicating the confidence we

have in the assumed p value.

Hyperparameter Model

- This is the distribution(s) used to model the

breadth of allowable values for the

hyperparameters themselves. - In our pass/fail example the hyperparameters were

p and Nm. But I only used a hyperprior for Nm

just for illustration purposes, e.g. gamma

distribution Nm GAM( h, d) - Solution requires numerical techniques that will

be discussed in Part-2 of these lectures.

References

- Bayesian Reliability, by Hamada, Wilson, Reese

and Martz, Wiley (2007) - Bayesian Reliability Analysis, by Martz Waller,

Wiley (1995) - Hierarchical MCMC for Bayesian System

Reliability, Los Alamos Report eqr094, June

2006. - Statistical Methods for Reliability Data by

Meeker Escobar, Wiley, (1998), Chapter 3

Appendices

Definitions etc.

Gamma and Beta Functions

- The complete beta function is defined by
- The gamma function is defined by

Classical Reliability

Bayes Theorem Applied to Reliability as measured

by Pass/Fail tests.

- Assume we are to perform a battery of tests the

result of each test is either a pass (x0) or a

fail (x1). - Traditionally we would run say n tests and record

how many passed, s, and then calculate the

proportion that passed s/n and label this as the

reliability. i.e. ltRgts/n. This is called a point

estimate of the reliability. - Then we would use this value of R to calculate

the probability of say s future successes in n

future tests, using the binomial probability

distribution p(sn,ltRgt)nCs

(ltRgt)s(1-ltRgt)(n-s) - This formulation assumes R is some fixed but

unknown constant that is estimated by the most

recent pass/fail data e.g. ltRgt. - ltRgt is bounded (Clopper-Pearson Bounds) see next

set of charts.

Classical Reliability analysis

- Classical approach to Pass/Fail reliability

analysis - To find a confidence interval for R using any

estimator e.g. ltRgt, one needs to know the

statistical distribution for the estimator. In

this case it is the binomial distribution.

Classical Reliability

- For example we know that if the number of tests

was fairly large that the binomial distribution

(the sample distribution for ltRgt) can be

approximated by a normal distribution whose mean

ltRgt and whose standard deviation, called the

standard error of the mean SEM (ltRgt(1-ltRgt)/n)1/2

and therefore the 1-sided confidence interval for

R is written out in terms of a probability

statement as follows - As an example suppose there were n6 tests

(almost too small a number to use a normal

approximation, and s5 successes, ltRgt5/6, take

a0.05 for a 95 lower reliability bound, Z1-a

1.645, SEM 0.152 so one has - This is a very wide interval as there were very

few tests.

Classical Reliability

- This is a fairly wide confidence interval which

is to be expected with so few tests. The

interval can only be made narrower by performing

more tests (increase n and hopefully s) or

reducing the confidence level from say 95 to say

80. Running the above calculation at a0.2

gives - This is the standard (frequentist) reliability

approach. Usually folks leave out the confidence

interval because it looks so bad when the number

of tests is low. - Actually n6 tests does not qualify as a very

large sample for a binomial distribution. In

fact of we perform an exact calculation using the

cumulative binomial distribution (using the RMS

tool SSE.xls) one finds for a confidence level

of 95. At a confidence level of 80 - These non parametric exact values give

conservatively large (but more accurate) answers

when the number of tests is small. - The expression for the nonparametric confidence

interval can be found using the RMS tools that

are used to compute sample size (e.g. SSE.xlsm

whose snapshot is shown below). Tools available

for download from eRoom at http//ds.rms.ray.com/d

s/dsweb/View/Collection-102393 look in excel

files for SSE.xlsm.

Raytheon Tools

"if all you have is a hammer, everything looks

like a nail. Abraham Maslow, 1966

Sample Size Estimator, SSE.xlsm

In SSE the gold bar can be shifted under any of

the 4 boxes (Trials, Max Failures, Prob of Test

Failure, Confidence Level) by double clicking

mouse button in cell below the number you are

trying to find. The gold bar indicates what will

be calculated. You will need to allow macro

operation in excel in order for this calculator

to work. Another choice would be the Exact

Confidence Interval for Classical Reliability

calculation from binary data excel spreadsheet

which is available from the author by email

request. The actual equations that are

calculated can be found in reference by Meeker

Escobar 4 or an alternative form can be copied

from the formulas below. The two sided bounds on

the reliability confidence interval for a

confidence level 1-a are given by

RLower(a,n,s) BETAINV(1-a/2,s,n-s1) and

RUpper(a,n,s) BETAINV(a/2,s1,n-s) where n

tests, ssuccesses in n tests, and CLconfidence

level1-a. The function BETAINV is in excel. For

a one sided calculation which is applicable for

ns (x0) calculations one can use a instead of

a/2 in RLower equation.

Exact 2-sided Confidence Interval

The yellow inserts show the binomial calculation

needed to evaluate the confidence bounds but

these binomial sums can be rewritten either in

terms of Inverse Beta distributions or Inverse F

distributions. Stated as a probability form one

has

Bayesian Search Algorithm

An example of how Bayesian techniques aided in

the search for a lost submarine

- View Backup Charts on Search for Submarine

Backup ChartsBayes Scorpion Example

- From Cressie and Wikle (2011) Statistics for

Spatio-Temporal Data

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