introduction to bayesian statistics

Introduction to Bayesian Statistics

Alyson Wilson, Ph.D., PStat R©

Department of StatisticsLaboratory for Analytic SciencesNorth Carolina State University

agwilso2@ncsu.edu

April 4, 2017

A. Wilson (NCSU Statistics) Introduction to Bayesian Statistics April 4, 2017 1 / 55

Background and InterestsBackground and Research Interests

A. Wilson (NCSU Statistics) Bayesian Analysis January 31, 2017 2 / 52A. Wilson (NCSU Statistics) Introduction to Bayesian Statistics April 4, 2017 2 / 55

When be Bayesian?

From Meeker and Hong (2014), “Many of the applications described inthis article and particularly in this concluding section will requirecombining information from different sources (e.g., data, inexactphysics-based knowledge, and certain kinds of expert opinion).Additionally, statistical models being used will often contain multiplesources of variability. Bayesian statistical methods provide a naturalapproach for combining such information.”

W. Q. Meeker and Y. Hong (2014). Reliability Meets Big Data:Opportunities and Challenges. Quality Engineering 26(1): 102 - 116.

Bayesian ReliabilityHeterogeneous Information

• DoD systems experience system design, contractor testing,developmental testing, and operational testing.

• Later in the lifecycle, we see new variants of systems, life extensionprograms, . . . .

• The goal of science-based stockpile stewardship is the assessment ofsafety and reliability in aging warheads in the absence of nucleartesting.

From these examples

• Common threads• We need to combine many sources of information• We need to use the information we have already seen to plan

subsequent data collection• Typically more broadly than factors, experimental region, and tentative

model structure

• Meeker and Hong also discuss extrapolation, a common desire inreliability analyses: “Extrapolation will be more reliable if predictionsare based on a combination of science-based models of reliability (e.g.,knowledge of the physics of well-understood failure modes) and dataare used to develop predictive models for a failure time distribution.”

Why be Bayesian?

• Allows incorporation of prior information• May supplement limited data• May provide improvements in cost or precision• Provides a formal framework to think about how to combine

information

• Computational simplifications• Censored data• When framed as a Bayesian problem, complex models can often be fit

relatively easily using Markov chain Monte Carlo or othercomputational algorithms.

• Straightforward to produce estimates and credible intervals forcomplicated functions of model parameters (e.g., predictions,probability of failure, quantiles of lifetime distribution)

• Philosophical Pragmatic

Example

Question of Interest: How likely is the upcoming launch by country X tobe successful?

Basic Statistics Problem: Unknown population parameter (θ) must beestimated.

θ = Probability of a successful launch of a new vehicle by an inexperiencedagent. During the period 1980–2002, eleven launches of new vehicles wereperformed by companies or agencies with little launch vehicle designexperience. Of these eleven, three were successful and eight were failures.

Launch Vehicle Outcome Data

Vehicle Outcome

Pegasus SuccessPercheron FailureAMROC FailureConestoga FailureAriane 1 SuccessIndia SLV-3 FailureIndia ASLV FailureIndia PSLV FailureShavit SuccessTaepodong FailureBrazil VLS Failure

ExampleStep 1: Careful Modeling

Step 1 of both the Bayesian and non-Bayesian formulations is to choose astatistical model (sampling distribution) for the data.

One choice for f (y | θ) is that the number of successful launches (Y )follows a binomial distribution with n launches and the probability of anyone test being a “success” denoted as θ.

Equivalently, we can think about each launch as a Bernoulli trial withprobability of θ.

What are some of the problems with this model?

For a More Detailed Analysis

V. Johnson, A. Moosman, P. Cotter (2005). A hierarchical model forestimating the early reliability of complex systems. IEEE Transactions onReliability 54(2): 224-231.

Classical/Frequentist (non-Bayesian) Analysis

1 All pertinent information enters the problem through the likelihoodfunction in the form of data (Y1, . . . ,Yn)

L(θ) = f (y | θ)

)θy (1− θ)n−y

2 Software packages all have this capability

3 Maximum likelihood, unbiased estimation, etc.

4 Confidence intervals, tests of hypotheses

Classical/Frequentist AnalysisMaximum Likelihood

log f (y | θ) ∝ y log(θ) + (n − y) log(1− θ),where y = 3, n = 11

Taking the first derivative of the log-likelihood with respect to θ andsetting the result equal to 0, we see that the MLE must satisfy

0 =∂ log f (y | θ)

∂θ=

θ− n − y

1− θ

Solving for θ implies that the MLE of θ, say θ̂, is given by

θ̂ =y

In other words, the MLE of the success probability θ in a binomial model issimply the observed proportion of successes.

Classical/Frequentist AnalysisMaximum Likelihood

Classical/Frequentist InferenceConfidence Intervals

For the launch vehicle data, a point estimate of the success probability ofa new launch system developed by an inexperienced manufacturer isprovided by the MLE:

θ̂ =y

11= 0.272

An interval estimate for the population proportion of success can beobtained using the asymptotic normal sampling distribution of the MLE θ̂.

For θ, we have a 90% confidence interval of

(0.272− 1.645× 0.134, 0.272 + 1.645× 0.134) = (0.052, 0.492)

In repeated sampling, one expects the confidence interval to include theunknown parameter with probability close to 0.90.

Bayesian AnalysisOne more piece of the model

1 Data enters through the likelihood function

2 However, other information can be incorporated through the priordistribution

3 Prior distribution: Before any data collection, the view of/informationabout the parameter

• Expressed as a probability distribution on θ• Can come from expert opinion, historical studies, previous research,

computer models, similar systems, subject matter knowledge, . . .• There exist “noninformative” (“flat,” “diffuse”) priors that represent

states of ignorance.

Bayesian Analysis

4 Bayes’ Theorem:

p(θ | y1, . . . , yn) =f (y | θ)× π(θ)∫f (y | θ)× π(θ)dθ

The posterior distribution, p(θ | bfy), is a constant multiplied by thelikelihood, f (y | θ), muliplied by the prior distribution, π(θ). (Theposterior distribution is proportional to the prior times the likelihood.)

5 Posterior distribution: In light of the data, the updated viewof/information about the parameter

6 All inference is based on the posterior distribution.

Thought Experiment

The Strength of Evidence versus the Power of Belief: Are We AllBayesians? Lecture by Jessica Utts at the International Conference onTeaching Statistics (2010).http://videolectures.net/icots2010_utts_awab/

• Parapsychology (“ESP”)

• Remote viewing experiment

• n = 2124 experiments, with y = 709 “successes,” where you wouldexpect a 25% success rate by chance alone

• The point estimate is 709/2124 = 0.334, with a 95% confidenceinterval of (0.314, 0.354).

• Are you convinced that the probability is greater than 0.25?

Thought Experiment

Suppose that you are a skeptic. Most likely value is 0.25, and you are 95%sure that the real value is less than 0.255.

Thought ExperimentSuppose that you are a believer. Most likely value is 0.33, and you are95% sure that the real value is below 0.36. (If this were a huge effect,we’d all know.)

Thought Experiment

Suppose that you are “open-minded.” Most likely value is 0.25, and youare 95% sure that the real value is below 0.30.

Thought Experiment

Priors Posteriors

Bayesian Analysis

1 “Subjective” (what/whose information is contained in the priordistribution?)

• Careful, detailed elicitation• Sensitivity analysis (These kinds of analyses are necessarily very explicit

about assumptions and very diligent in understanding the impact ofeach assumption and source of information on inference and decisions.)

• Calibration of priors• Experts, historical data, previous experiments, computer models,

subcomponent tests

2 Increasing number of software packages have this capability (SASPROC MCMC, OpenBUGS, JAGS, STAN, NIMBLE, R packages)

3 Result is a probability distribution

4 Credible intervals use the language that everyone wants to use.(Probability that θ is in the interval is 0.90.)

ExampleBayesian Analysis: Beta-Binomial

A convenient choice to represent prior information about the probability ofa successful launch is the Beta distribution. One interpretation of theparameters defining this distribution are the number of a priori successesand failures.

For example, if an expert hypothesizes that her opinion about theprobability of successful launch is worth 8 vehicle launches vehicles andfurther expects successful launches 6 times, we would reflect this with aBeta(6, 2) distribution.

Example

• Non-Bayesian analysis: If our data are Binomial(n, θ) then we wouldcalculate Y /n as our estimate and use a confidence interval formulafor a proportion.

• Bayesian analysis: If our data are Binomial(n, θ) and our priordistribution is Beta(a, b), then our posterior distribution isBeta(a + y , b + n − y).

ExampleBayesian Analysis

• In this case, the posterior distribution isBeta(6 + 3, 2 + 11− 3) = Beta(9, 10).

• This means that we can say that the probability that θ is in theinterval (0.26, 0.69) is 0.95.

• Notice that we don’t have to address the problem of “in repeatedsampling”; this is a direct probability statement that relies on theprior distribution.

ExampleBaseline Posterior Distribution

n = 11, y = 3, a = 6, b = 2

0.0 0.2 0.4 0.6 0.8 1.0

Proportion

ExampleDiffuse Prior Posterior Distribution

n = 11, y = 3, a = 1, b = 1

0.0 0.2 0.4 0.6 0.8 1.0

Proportion

ExampleLarge n Baseline Posterior Distribution

n = 110, y = 30, a = 6, b = 2

0.0 0.2 0.4 0.6 0.8 1.0

Proportion

ExampleLarge n Diffuse Prior Posterior Distribution

n = 110, y = 30, a = 1, b = 1

0.0 0.2 0.4 0.6 0.8 1.0

Proportion

ExampleAll Four Posterior Distributions

0.0 0.2 0.4 0.6 0.8 1.0

Proportion

Beta(6,2), n=11Beta(1,1), n=11Beta(6,2), n=110Beta(1,1), n=110

Credible Intervals

90% confidence interval: (0.052, 0.492)

90% credible intervals

• Beta(6,2), n = 11: (0.291,0.659)

• Uniform(0,1), n = 11: (0.123,0.527)

• Beta(6,2), n = 110: (0.238,0.376)

• Uniform(0,1), n = 110: (0.210,0.348)

ExampleConsider an example involving continuous-valued random variables. Theparticular data set we consider represents viscosity breakdown times for 50samples of a lubricant.

Example

Question of Interest: What is the time of viscosity breakdown for aparticular lubricating fluid?

Basic Statistics Problem: Unknown population parameters (θ) must beestimated.

θ = Mean and standard deviation of (log) viscosity breakdown times for aparticular lubricating fluid. Data are collected (in 1000s of hours) for 50samples.

Viscosity Breakdown Times

Viscosity breakdown times (in 1000s of hours) for 50 samples of alubricating fluid.

5.45 16.46 15.70 10.39 6.71 3.77 7.42 6.899.45 5.89 7.39 5.61 16.55 12.63 8.18 10.446.03 13.96 5.19 10.96 14.73 6.21 5.69 8.184.49 3.71 5.84 10.97 6.81 10.16 4.34 9.814.30 8.91 10.07 5.85 4.95 7.30 4.81 8.446.56 9.40 11.29 12.04 1.24 3.45 11.28 6.645.74 6.79

ExampleStep 1: Careful Modeling

Step 1 of both the Bayesian and non-Bayesian formulations is to choose astatistical model (sampling distribution) for the data.

One choice for f (y | θ) is that the natural logarithm of the viscositybreakdown times (log(Y )) has an normal distribution with parametersθ = (µ, σ2).

Specifying the Model

• θ = (µ, σ2) are the mean and variance of the natural logarithm of thelubricant measurements.

• A convenient choice for π(θ) is a normal distribution for µ and aninverse gamma distribution for σ2. We’ll assume mutualindependence.

• System engineers who have studied the viscous properties of theselubricants believe that log(Y ) should be centered at 2, but that valueis known only with standard deviation 1.

• They have very little knowledge about the variability they expect tosee.

• The process of getting a prior distribution from statements made byexperts is called elicitation.

• How could we check this prior distribution with the engineers?

Joint Posterior Distribution

Marginal Posterior Distributions

ComputationA Brief Mention

• What do we do when the posterior distribution of θ does not have anice distributional form?

• Note that the dimension of θ is often large in real problems.

• Direct numerical integration (for example, to determine means ormarginal distributions) is problematic

• We could construct a normal approximation . . .

• The answer is: We sample. In particular, we figure out how to draw arandom sample from p(θ | y1, . . . , yn). Then we can computequantities of interest using Monte Carlo techniques.

Computation

There are a variety of algorithms that can be used to get our randomsamples:

• Rejection sampling

• Importance sampling

• Sampling importance resampling

• Gibbs sampling

• Metropolis-Hastings algorithm

System Reliability

Statistical Areas of InterestSystem Reliability

• Data• Multilevel• Multiple types: binary, lifetime,

degradation, expert judgement,computer model

• Systems• Representation• Assessment: model checking and

diagnostics, model fit

• Planning data collection

System Reliability

Multilevel DataMultilevel Pass/Fail Data, Series System

• Information collected at C0, C1, C2, andC3

• Information at C0 provides partialinformation about C1, C2, and C3

• Goal: simultaneous inference aboutsystem and component reliabilities

Successes Failures Trials

Component 1 8 2 10

Component 2 7 2 9

Component 3 3 1 4

System 10 2 12

System Reliability

Example: System ReliabilityMultilevel Pass/Fail Data, Series System

C1 C2 C3

UnitsSuccesses Failures Tested

Component 1 8 2 10Component 2 7 2 9Component 3 3 1 4System 10 2 12

L(p1, p2, p3; x) = p81(1− p1)2p72(1− p2)2p33(1− p3)p10S (1− pS)2

= p81(1− p1)2p72(1− p2)2p33(1− p3)(p1p2p3)10(1− p1p2p3)2

Prior information for p1, p2, p3, pS

System Reliability

C1 C2 C3

= p81(1− p1)2p72(1− p2)2p33(1− p3)(p1p2p3)10(1− p1p2p3)2

System Reliability

C1 C2 C3

= p81(1− p1)2p72(1− p2)2p33(1− p3)(p1p2p3)10(1− p1p2p3)2

System Reliability

System Representations

Reliability Block Diagram

Fault Tree

Bayesian Network

System Reliability

ExampleMultilevel Pass/Fail Data, Bayesian Network, Reliability Changing with Time

��C

��A

��B

Age A B C

1 19/19 35/35 15/162 - 47/48 14/143 16/19 37/38 12/144 12/12 - -5 - 44/45 13/146 - 35/37 11/127 9/13 - -8 - 33/42 5/169 - - 12/1910 3/10 30/39 8/14

System Reliability

System RepresentationsDeveloping the model for a complex system

System Reliability

Extensions

This basic approach has been extended in many directions.

• Multiple diagnostics measured at the components (Anderson-Cook etal. (2008))

• Binary, lifetime, or degradation data at components and system (Guoand Wilson (2013))

• Parallel line of research focused on developing system models:elicitation, software, representations, model checking (e.g., Wilson etal. (2007), Anderson-Cook (2008), Zhang and Wilson (2016))

• Prior distributions (to capture knowledge about parameters “before”this experiment)

• “Naive” specifications can lead to surprisingly bad results• Variable selection priors

System Reliability

Favorite Books

Popular Science

• S. McGrayne (2011). The Theory that Would Not Die: How Bayes’Rule Cracked the Enigma Code, Hunted Down Russian Submarines,and Emerged Triumphant from Two Centuries of Controversy. YaleUniversity Press.

Bayesian Reliability

• H. Martz, R. Waller (1982). Bayesian Reliability Analysis. John Wiley& Sons.

• N. Singpurwalla (2006). Reliability and Risk: A Bayesian Perspective.Wiley.

• M. Hamada, A. Wilson, C. S. Reese, H. Martz (2008). BayesianReliability. Springer.

System Reliability

Favorite Books

Introductory Bayesian Methods

• B. Carlin, T. Louis (2008). Bayesian Methods for Data Analysis, 3rdEdition. Chapman and Hall.

• P. Hoff (2009). A First Course in Bayesian Statistical Methods.Springer.

Eliciting Prior Distributions

• M. Meyer, J. Booker (2001). Eliciting and Analyzing ExpertJudgment: A Practical Guide. ASA/SIAM.

• A. O’Hagan et al. (2006). Uncertain Judgements: Eliciting Experts’Probabilities. Wiley.

System Reliability

Favorite Books

Bayesian Computing

• D. Gamerman, H. Lopes (2006). Markov Chain Monte Carlo:Stochastic Simulation for Bayesian Inference, 2nd Edition. Chapman& Hall/CRC.

• J. Albert (2009). Bayesian Computation with R, 2nd edition.Springer.

• C. Robert, G. Casella (2010). Introducting Monte Carlo Methodswith R. Springer.

• C. Robert, G. Casella (2010). Monte Carlo Statistical Methods.Springer.

System Reliability

Bayesian Reliability Articles I

• C. Anderson-Cook (2008). Evaluating the Series or Parallel StructureAssumption for System Reliability. Quality Engineering 21(1): 88-95.

• C. M. Anderson-Cook, S. Crowder, A. V. Huzurbazar, J. Lorio, J.Ringland, A. G. Wilson (2011). Quantifying reliability uncertaintyfrom catastrophic and margin defects: A proof of concept. ReliabilityEngineering and System Safety 96: 1063-1075.

• C. Anderson-Cook, T. Graves, M. Hamada, N. Hengartner, V.Johnson, C. S. Reese, A. Wilson (2007). Bayesian StockpileReliability Methodology for Complex Systems with Application to aMunitions Stockpile. Journal of the Military Operations ResearchSociety 12(2): 25-38.

• C. Anderson-Cook, T. Graves, N. Hengartner, R. Klamann, A.Koehler, A. Wilson, G. Anderson, G. Lopez (2008). ReliabilityModeling Using Both System Test and Quality Assurance Data.Journal of the Military Operations Research Society 13: 5-18.

System Reliability

Bayesian Reliability Articles II• J. Chapman, M. Morris, C. Anderson-Cook (2012). Computationally

Efficient Comparison of Experimental Designs for System ReliabilityStudies with Binomial Data. Technometrics 54(4): 410-424.

• T. Graves, M. Hamada, R. Klamann, A. Koehler, H. Martz (2008).Using simultaneous higher-level and partial lower-level data inreliability assessments. Reliability Engineering and System Safety93(8): 12731279.

• J. Guo, A. Wilson (2013). Bayesian Methods for Estimating theReliability of Complex Systems using Heterogeneous MultilevelInformation. Technometrics 55(4): 461-472.

• M. Hamada, H. F. Martz, C. S. Reese, T. Graves, V. Johnson, A. G.Wilson (2004). A fully Bayesian approach for combining multilevelfailure information in fault tree quantification and optimal follow-onresource allocation. Reliability Engineering and System Safety 86:297-305.

System Reliability

Bayesian Reliability Articles III• M. Hamada, A. Wilson, B. Weaver, R. Griffiths, H. Martz (2014).

Bayesian Binomial Assurance Tests for System Reliability UsingComponent Data. Journal of Quality Technology 46(1): 24-32.

• V. Johnson, T. Graves, M. Hamada, and C. S. Reese (2003). Ahierarchical model for estimating the reliability of complex systems. InBayesian Statistics 7, eds. Bernardo, J., Bayarri, M., Berger, J.,David, A., Heckerman, D., Smith, A., and West, M., OxfordUniversity Press, pp. 199-213.

• V. Johnson, A. Moosman, P. Cotter (2005). A Hierarchical Model forEstimating the Early Reliability of Complex Systems. IEEETransactions on Reliability 54(2): 224-231.

• M. Li, W. Meeker (2014). Application of Bayesian Methods inReliability Data Analyses. Journal of Quality Technology 46(1): 1-23.

• W. Meeker, Y. Hong (2014). Reliability Meets Big Data:Opportunities and Challenges. Quality Engineering 26(1): 102-116.

System Reliability

Bayesian Reliability Articles IV• C. S. Reese, A. Wilson, M. Hamada, H. Martz, K. Ryan (2004).

Integrated Analysis of Computer and Physical Experiments.Technometrics 46(2): 153-164.

• C. S. Reese, A. Wilson, J. Guo, M. Hamada, V. Johnson (2011). AHierarchical Model for Estimating Reliability from Weapon SystemSurveillance Data. Journal of Quality Technology 43(2): 127-141.

• A. Wilson, C. Anderson-Cook, A. Huzurbazar (2011). A Case Studyfor Quantifying System Reliability and Uncertainty. ReliabilityEngineering and System Safety, 96(9): 1076-1084.

• A. Wilson, T. Graves, M. Hamada, C. S. Reese (2006). Advances inData Combination, Analysis, and Collection for System ReliabilityAssessment. Statistical Science 21(4): 514-531.

• A. Wilson, A. Huzurbazar (2007). Bayesian Networks for MultilevelSystem Reliability. Reliability Engineering and Systems Safety 92(10):1413-1420.

System Reliability

Bayesian Reliability Articles V

• A. Wilson, L. McNamara, G. Wilson (2007). Information integrationfor complex systems. Reliability Engineering and System Safety 92:121–130.

• X. Zhang, A. Wilson (2016). System Reliability and ComponentImportance under Dependence: A Copula Approach. To appear inTechnometrics.

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