Bayesian Statistics - PowerPoint Presentation

Slide1

Bayesian Statistics

:Applied to Reliability: Part 1 Rev. 1

Allan Mense, Ph.D., PE, CREPrincipal Engineering FellowRaytheon Missile SystemsTucson, AZ

1Slide2

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.2Slide3

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 EscobarRev. 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.

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Bayesian Helps

4

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.Slide5

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 byP(A and B)=P(A|B)P(B) = P(B and A)=P(B|A)P(A)=P(AB)There is no implication of time series of these eventsP{A|B) is called a conditional probabilityP(A and B)= P(A)P(B) if A and B are independent, P(A and B)=0 if A and B are mutually exclusiveExample: Have 3 subsystems in series the probability the systems does not fail by time T is RSYS=R1(T)*R2(T)*R3(T) if failures are not correlated5

These rules also work for probability distributionsSlide6

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 byP(A or B)=P(A)+P(B) –P(A and B) = P(AB)There is no implication of time series of these eventsP(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 exclusiveExample: Have 2 subsystems in parallel the probability the systems does not fail by time T is RSYS=R1(T)+R2(T)-R1(T)R2(T) if failures are not correlated6

A

B

A

B

These rules also work for probability distributionsSlide7

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 byP(A | B)=P(B|A) P(A) / P(B)There is no implication of time series of these events.P(A|B)= posterior probability. P(A) = prior probability. P(B)=P(B|A)P(A)+P(B|~A) P(~A), called the marginal probability of event B also called the “Rule of Total Probability.” 7These rules also work for probability distributionsSlide8

Problem #1

A=event that missile thruster #1 will not work=0.15

B=event that missile thruster #2 will not work= 0.15P(A or B or both A and B) = P(A)+P(B)-P(A)P(B) = 0.15+0.15-(0.15)(0.15) = 0.30-0.0225 = 0.2775P(A or B but not both A and B)= P(A)+P(B)-2P(A)P(B) = 0.15 + 0.15 – 0.045 = 0.2558Slide9

Problem #2

Let A = man age > 50 has prostate cancer,

B = PSA score >5.0, 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(B|A)=0.9, P(B|~A) = 0.2 i.e. even if you do not have cancer the test registers PSA>5.0 about 20% of the time (false positive), Thus the probability of getting a PSA score > 5.0 is given by P(B)=P(B|A)P(A)+P(B|~A)P(~A) = (0.9)(0.3)+(0.2)(0.7)=0.41Using Bayes’ Theorem P(A|B)= P(B|A)P(A)/P(B)=(0.9)(0.3)/0.41=0.66Your probability of having prostate cancer has gone from 30% with no knowledge of test to 66% given your knowledge that your PSA > 5.09Slide10

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(s|n,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. (R|n,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| N

m

,

p

) ~ R

Nm

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. N

m

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)

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Now we need to study each part of this equation starting with the likelihoodSlide11

Binomial Distribution for the Data

Probability of an event occurring = p

Probability of event not occurring = 1-pProbability that event occurs x=2 times in n=5 tests1st 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)3In 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!)11Slide12

Binomial distribution

Since any of the 10 instantiations would give the same outcome (i.e. 2 successes in 5 trials each with probability p

2(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(x|n, p)= n

Cx p

x(1-p)(n-x)The

random variable is x, the number of successful events

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binomial distribution

random variable is x, the number of successes

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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(p|x, n)= likelihood that the probability of a successful event is p given the data x and n.L(p|x, n) ≡ px (1-p

)(n-x)The random variable is now p

with x and n known14

The Likelihood function is put to great use in Bayesian statisticsSlide15

Likelihood Function

, L(p|s,n)The random variable is p, the probability of a success

15Sometimes we label p as R, the reliabilitySlide16

Likelihood Functions

*: What they look like!

161 Test and 1 Success1 Test and 0 Successes

*

Likelihood function normalized to area under function = 1. This is not necessary but allows for a better interpretationSlide17

Likelihood Functions for n=2 tests

17

0 successes

2 successes

1 successSlide18

Bayesian Approach

Use Bayes’ Theorem with probability density functions e.g.

P(A|B) => fposterior(parameter|data) P(B|A) => Likelihood(data| parameter) P(A) => fprior(parameter before take data)fposterior  Likelihood X f

priorSteps: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.

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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.19Slide20

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 testsPass/Fail testsTime dependent tests (exponential & Weibull)Counting experiments (Poisson statistics)20Slide21

Bayes’ Theorem for Distributions

when using pass/fail data

fposterior(R|n, s, info)=L(s, n|R) X fprior(R| info)Example: For pass/fail datafprior(R)  Ra-1(1-R)b-1 (beta)L(s,n|R)  Rs(1-R)n-s (binomial)fposterior(R| s, n, a, b

)  Rs+

a-1(1-R)n-s+b

-1 (beta-binomial)

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Much more will be said of this formulation laterSlide22

Prior

, Likelihood, Posterior

What they look like! 22You want the information that comes from knowing the green curve!Slide23

Bayesian Calculator

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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.Slide24

Bayesian gives lower probabilities for high success and higher values for low success.

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Bayesian keeps one from being too pessimistic or too optimisticSlide25

Formulas Used In Beta-binomial model for pass/fail data

Prior Distribution:Likelihood:

Posterior:25

The prior uses what I call the Los Alamos parameterization of the beta distribution. See Appendix for more detailSlide26

p

=0.9, Nm=10

26Take # tests n=4 and resulting successes s=3Note 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 R=0.857 where a classical calculation would give <R>=3/4=0.75.Slide27

p

=0.9, Nm=2

27Take # tests n=4 and resulting successes s=3Note how weak prior distribution (Nm=2) will still pull the posterior up that its net effect is much smaller i.e. the posterior has a maximum value at R=0.800 where a classical calculation would give <R>=3/4=0.75. So the posterior and likelihood are more closely aligned.Slide28

p

=0.9, Nm=10

28Increase tests: Take # tests n=20 and resulting successes s=15Note how strong prior distribution still pulls the posterior up but due to the larger amount of data (n=20) the posterior distribution still shows a maximum value at R=0.800 so the prior is beginning to be overpowered by the data.Slide29

p

=0.9, Nm=2

29Take # tests n=20 and resulting successes s=15Note how weak prior distribution still has some effect but due to the larger amount of data (n=20) the posterior distribution is almost coincident with the likelihood (i.e. the data) and shows a maximum value at R=0.764 which is close to the data modal value of 0.75. The data is overpowering the prior.Slide30

N

m=0 uniform prior

30Take # tests n=20 and resulting successes s=15Note 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 R=0.750 . The data is all we have. Note that p plays no role because all values of R are equally likely when Nm=0. 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.Slide31

Look at Predictions i.e. # successes in the next 10 tests

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p=0.9, Nm=10, n=20, s=15, <R>=15/20=0.75

p

=0.9, Nm

=10, n=4, s=3, <R>=3/4=0.75

The Bayesian calculation shows that due to the strong prior (N

m

=10) the prediction of success is higher for large numbers of successes as compared to the use of the average reliability in the binomial calculationSlide32

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 knowledgeDepends 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? YESWhat if we have uncertainty in N

m?Set up a hyperprior distribution for N

m – say a gamma distribution. The parameters of f(N

m) i.e. (h,d) are called hyperhyperparameters.

How do you choose the parameters

d

and

h

for the gamma distribution? Pick mode for N

m

=

(d-1)/h

and stdev of N

m

=

d

1/2

/ h

.

Example: for a weak prior we might choose the mode = 3 for N

m

, and a stdev =2, which gives

h=1, d=4

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Now it gets complicated!

Up until now we have been able to solve for f

posterior(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 belowThere 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.33

Will apply MCMC to find

f

posterior

(R), learn this in part-2 of lecturesSlide34

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 solution34Numerical Techniques will be the subject of Part – 2 of these lecturesSlide35

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 l

T? How do we handle that variability?

35

Applying a prior distribution to the parameter

l

is the Bayesian processSlide36

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, i=1,2,…,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(-l

tr)] X [exp(-l

tr+1) exp(-

l

t

r+2

)…

exp(-

l

t

n

)]

=

l

r

exp(-

l

(t

1

+t

2

+…+t

r

+ (n-r)t

R

))=

l

r

exp(-

l(

TTT)), TTT=total 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,

l

MLE

= 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 R=exp(-

l

t

). So for any given time say t=t1 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.

36

Applying a prior distribution to the parameter

l

is the Bayesian processSlide37

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 37Slide38

Exponential time-to-failure Distribution

Model variation of l

with Gamma distributionMultiplying Prior X Likelihood givesSo 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(n+a, b+TTT), TTT=(t1+t2+…+tn)=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.

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R(t)

To find the reliability as a function of time one must integrate over l

from 0 to ∞, 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.39

This will be evaluated later. Slide40

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 let’s 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.

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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.41Slide42

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(y|q) e.g. f(s|R) = binomial distribution, and the prior distribution as g(q|a) where a represents the parameters of the prior distribution (often called hyperparameters) e.g. g(R|Nm

,p) = beta distribution. It

may be the case that we know g(q|

a) 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(N

m

) since we know

p

but do not know N

m

.

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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.(

q|a) ̴ g

(q|

a

)

3

. The

hyperparameter model

for the parameters of the structural model

.

a

̴

h(

a

)

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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(t;a,b) = exp(-(t/a)b), which is the result we are seeking.

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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(R|p,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 = N

mp+1, b = N

m(1-p)+1), because

p

= mode of the distribution and N

m

is a weighting factor indicating the confidence we have in the assumed

p

value.

45Slide46

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.46Slide47

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 347Slide48

Appendices

48Slide49

Definitions etc.

49Slide50

Gamma and Beta Functions

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

50Slide51

Classical Reliability

51Slide52

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 (x=0) or a fail (x=1).

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. <R>=s/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(s*|n*,<R>)=n*Cs* (<R>)s*(1-<R>)(n*-s*)This formulation assumes R is some fixed but unknown constant that is estimated by the most recent pass/fail data e.g. <R>. <R> is bounded (Clopper-Pearson Bounds) see next set of charts.52Slide53

Classical Reliability analysis

Classical approach to Pass/Fail reliability analysisTo find a confidence interval for R using any estimator e.g. <R>, one needs to know the statistical distribution for the estimator. In this case it is the binomial distribution.

53Slide54

Classical Reliability

For example we know that if the number of tests was fairly large that the binomial distribution (the sample distribution for <R>) can be approximated by a normal distribution whose mean = <R> and whose standard deviation, called the standard error of the mean SEM= (<R>(1-<R>)/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 n=6 tests (almost too small a number to use a normal approximation, and s=5 successes, <R>=5/6, take a=0.05 for a 95% lower reliability bound, Z1-a = 1.645, SEM = 0.152 so one hasThis is a very wide interval as there were very few tests.54Slide55

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

a=0.2 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 n=6 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/ds/dsweb/View/Collection-102393 look in excel files for SSE.xlsm.55Slide56

Raytheon Tools

56

“"if all you have is a hammer, everything looks like a nail.” Abraham Maslow, 1966Slide57

Sample Size Estimator, SSE.xlsm

57

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-s+1) and R

Upper(a,n,s)= BETAINV(a/2,s+1,n-s) where n = # tests, s=#successes in n tests, and CL=confidence level=

1-a. The function BETAINV is in excel. For a one sided calculation which is applicable for n=s (x=0) calculations one can use a instead of

a/2

in R

Lower

equation.Slide58

Exact 2-sided Confidence Interval

58

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 hasSlide59

Bayesian Search Algorithm

View Backup Charts on Search for Submarine

59An example of how Bayesian techniques aided in the search for a lost submarineSlide60

Backup ChartsBayes Scorpion Example

From Cressie and Wikle (2011) Statistics for Spatio-Temporal Data

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