1、 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA/SP-2009-569 Bayesian Inference for NASA Probabilistic Risk and Reliability Analysis Dr. Homayoon Dezfuli NASA Project Manager, NASA Headquarters, Washington, DC Dana Kelly Idaho National Laborator
2、y, Idaho Falls, ID Dr. Curtis Smith Idaho National Laboratory, Idaho Falls, ID Kurt Vedros Idaho National Laboratory, Idaho Falls, ID William Galyean Idaho National Laboratory, Idaho Falls, ID National Aeronautics and Space Administration June 2009Provided by IHSNot for ResaleNo reproduction or netw
3、orking permitted without license from IHS-,-,-NASA STI Program . in Profile Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA scientific and technical information (STI) program plays a key part in helping NASA maintain this important role. The
4、NASA STI program operates under the auspices of the Agency Chief Information Officer. It collects, organizes, provides for archiving, and disseminates NASAs STI. The NASA STI program provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical Report Server
5、, thus providing one of the largest collections of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, which includes the following report types: TECHNICAL PUBLICATION. Reports of completed research or a major si
6、gnificant phase of research that present the results of NASA Programs and include extensive data or theoretical analysis. Includes compilations of significant scientific and technical data and information deemed to be of continuing reference value. NASA counterpart of peer-reviewed formal profession
7、al papers but has less stringent limitations on manuscript length and extent of graphic presentations. TECHNICAL MEMORANDUM. Scientific and technical findings that are preliminary or of specialized interest, e.g., quick release reports, working papers, and bibliographies that contain minimal annotat
8、ion. Does not contain extensive analysis. CONTRACTOR REPORT. Scientific and technical findings by NASA-sponsored contractors and grantees. CONFERENCE PUBLICATION. Collected papers from scientific and technical conferences, symposia, seminars, or other meetings sponsored or co-sponsored by NASA. SPEC
9、IAL PUBLICATION. Scientific, technical, or historical information from NASA programs, projects, and missions, often concerned with subjects having substantial public interest. TECHNICAL TRANSLATION. English-language translations of foreign scientific and technical material pertinent to NASAs mission
10、. Specialized services also include creating custom thesauri, building customized databases, and organizing and publishing research results. For more information about the NASA STI program, see the following: Access the NASA STI program home page at http:/www.sti.nasa.gov E-mail your question via th
11、e Internet to helpsti.nasa.gov Fax your question to the NASA STI Help Desk at 443-757-5803 Phone the NASA STI Help Desk at 443-757-5802 Write to: NASA STI Help Desk NASA Center for AeroSpace Information 7115 Standard Drive Hanover, MD 21076-1320 Provided by IHSNot for ResaleNo reproduction or networ
12、king permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I Foreword This NASA-HANDBOOK is published by the National Aeronautics and Space Administration (NASA) to provide a Bayesian foundation for framing probabilis
13、tic problems and performing inference on these problems. It is aimed at scientists and engineers and provides an analytical structure for combining data and information from various sources to generate estimates of the parameters of uncertainty distributions used in risk and reliability models. The
14、overall approach taken in this document is to give both a broad perspective on data analysis issues and a narrow focus on the methods required to implement a comprehensive database repository. It is intended for use across NASA. Recent years have seen significant advances in the use of risk analysis
15、 at NASA. These advances are reflected both in the state of practice of risk analysis within projects, and in the status of several NASA requirements and procedural documents. Because risk and reliability models are intended to support decision processes, it is critical that inference methods used i
16、n these models be robust and technically sound. To this end, the Office of Safety and Mission Assurance (OSMA) undertook the development of this document. This activity, along with other ongoing OSMA-sponsored projects related to risk and reliability, supports the attainment of the holistic and risk
17、-informed decision-making environment that NASA intends to adopt. This document is not intended to prescribe any technical procedure and/or software tool. The coverage of the technical topics is also limited with respect to (1) the historical genesis of Bayesian methods; (2) comparisons of “classica
18、l statistics” approaches with Bayesian ones; (3) the detailed mathematics of a particular method (unless needed to apply the method); and (5) a source of actual reliability or risk data/information. Additionally, this document is focused on hardware failures; excluded from the current scope are spec
19、ific inference approaches for phenomenological, software, and human failures. As with many disciplines, there are bound to be differences in technical and implementation approaches. The authors acknowledge that these differences exist and to the extent practical these instances have been identified.
20、 The Bayesian Inference handbook assumes that probabilistic inference problems range from simple, well-supported cases to complex, multi-dimensional problems. Consequently, the approaches provided to evaluate these diverse sets of issues range from single-line spreadsheet formula approaches to Monte
21、 Carlo-based sampling methods. As such, the level of analyst sophistication should be commensurate with the issue complexity and the selected computational engine. To assist analysts in applying the inference principles, the document provides “call out” boxes to provide definitions, warnings, and ti
22、ps. In addition, a hypothetical (but representative) system analysis and multiple examples are provided, as are methods to extend the analysis to accommodate real-world complications such as uncertain, censored, and disparate data. For most of the example problems, the Bayesian Inference handbook us
23、es a modern computational approach known as Markov chain Monte Carlo (MCMC). Salient references provide the technical basis and mechanics of MCMC approaches. MCMC methods work for simple cases, but more importantly, they work efficiently on very complex cases. Bayesian inference tends to become comp
24、utationally intensive when the analysis involves multiple parameters and correspondingly high-dimensional integration. MCMC methods were described in the early 1950s in research into Monte Carlo sampling at Los Alamos. Recently, with the advance of computing power and improved analysis algorithms, M
25、CMC is increasingly being used for a variety of Bayesian inference problems. MCMC is effectively (although not literally) numerical (Monte Carlo) integration by way of Markov chains. Inference is performed by sampling from a target distribution (i.e., a specially constructed Markov chain, based upon
26、 the inference problem) until convergence (to the posterior distribution) is achieved. The MCMC approach may be implemented using Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-BAYESIAN INFERENCE FOR NASA PROBABILISTIC RISK AND RELIABILITY ANALYSIS
27、IIcustom-written routines or existing general purpose commercial or open-source software. In the Bayesian Inference document, an open-source program called OpenBUGS (commonly referred to as WinBUGS) is used to solve the inference problems that are described. A powerful feature of OpenBUGS is its aut
28、omatic selection of an appropriate MCMC sampling scheme for a given problem. The approach that is taken in the document is to provide analysis “building blocks” that can be modified, combined, or used as-is to solve a variety of challenging problems. Not just for risk and reliability inference, MCMC
29、 methods are also being used for other U.S. Government activities (e.g., at the Food and Drug Administration, the Nuclear Regulatory Commission, National Institute of Standards and Technology, and the National Atmospheric Release Advisory Center). The MCMC approach used in the document is implemente
30、d via textual scripts similar to a macro-type programming language. Accompanying each script is a graphical diagram illustrating the elements of the script and the overall inference problem being solved. In a production environment, analysis could take place by running a script (with modifications k
31、eyed to the problem-specific information). Alternatively, other implementation approaches could include: (1) using an interface-driven front-end to automate an applicable script, (2) encapsulating an applicable script into a spreadsheet function call, or (3) creating an automated script-based infere
32、nce engine as part of an information management system. In lieu of using the suggested MCMC-based analysis approach, some of the documents inference problems could be solved using the underlying mathematics. However, this alternative numerical approach is limited because several of the inference pro
33、blems are difficult to solve either analytically or via traditional numerical methods. As a companion activity to this handbook, a data repository and information management system is being planned by OSMA. The approaches described in the report will be considered in the development of the inference
34、 engine for this system. While not a risk or reliability data source itself, the Bayesian Inference document describes typical sources of generic and NASA-specific data and information. Further, examples of failure taxonomies and associated hierarchies of information are discussed. Some of the examp
35、les and case studies use real data nonetheless the inference results produced in this document should not be used for analysis. Follow-up activities to this document include developing a stand-alone supporting document describing the technical detail of the methods described herein. It is important
36、to note that having the detailed mathematics is not required to run or understand the approaches described in the report. A limited set of the examples provided in this report have been validated, but the results and associated tools will be further checked for correctness via a more formal approach
37、. Additional examples will be introduced in future revisions. Comments and questions concerning the contents of this publication should be referred to the National Aeronautics and Space Administration, Director, Safety and Assurance Requirements Division, Office of Safety and Mission Assurance, Wash
38、ington, DC 20546. Dr. Homayoon Dezfuli Project Manager, NASA Headquarters June 2009 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-BAYESIAN INFERENCE FOR NASA PROBABILISTIC RISK AND RELIABILITY ANALYSIS III Contents Foreword i Tables . vi Figures. v
39、ii Scripts. xiii Examples. vxii Acknowledgements. xix Acronyms. xx 1. Introduction 1 1.1 Risk and Reliability Data Analysis Objectives . 1 1.2 Taxonomy of Uncertainty . 1 1.3 Data-Model Relationship 5 1.3.1 Modeling system performance . 5 1.3.2 How simple is tossing a coin? 7 1.3.3 What is data? 8 2
40、. Bayesian Inference 11 2.1 Introduction 11 2.2 A simple example Tossing a coin to collect data . 14 3. Data Collection Methods . 17 3.1 Introduction 17 3.1.1 Key Topical Areas . 17 3.1.1 Types of Information and Data . 17 3.2 Mission Development Phase and Data Availability 18 3.3 Sources of Data an
41、d Information . 19 3.3.1 NASA and Aerospace-Related Data Sources 19 3.3.2 Other Data Sources 19 3.4 Risk and Reliability Lexicon 20 3.5 Taxonomies and Classification 21 3.6 Case Study System Description 24 3.6.1 Active Thermal Control System (ATCS) 24 3.6.2 ATCS Component Taxonomy 25 4. Parameter Es
42、timation 27 4.1 Preface to Chapter 4 27 4.2 Inference for Common Aleatory Models 28 4.2.1 Binomial Distribution for Failures on Demand . 29 4.2.2 Poisson Distribution for Initiating Events or Failures in Time . 37 4.2.3 Exponential Distribution for Random Durations 42 4.2.4 Multinomial Distribution
43、45 4.2.5 Developing Prior Distributions 47 4.3 Model Validation . 57 4.3.1 Checking the Validity of the Binomial Model . 57 4.3.2 Checking the Validity of the Poisson Model 64 4.3.3 Checking the Validity of the Exponential Model 71 Provided by IHSNot for ResaleNo reproduction or networking permitted
44、 without license from IHS-,-,-BAYESIAN INFERENCE FOR NASA PROBABILISTIC RISK AND RELIABILITY ANALYSIS IV 4.4 Time-Trend Models for p and Lambda 75 4.4.1 Time-Trend Model for p in the Binomial Distribution . 75 4.4.2 Time-Trend Model for Lambda in the Poisson Distribution 80 4.5 Population Variabilit
45、y Models . 85 4.5.1 Population Variability Model for p in the Binomial Distribution 85 4.5.2 Population Variability Model for Lambda in the Poisson Distribution 91 4.5.3 Population Variability Models for CCF Parameters 95 4.6 Modeling Time-to-Failure or Other Durations 98 4.6.1 Alternatives to Expon
46、ential Distribution for Random Durations . 99 4.6.2 Choosing Among Alternative DistributionsDeviance Information Criterion . 104 4.7 Analyzing Failure Time Data from Repairable Systems . 106 4.7.1 Repair Same as NewRenewal Process 106 4.7.2 Repair Same as OldNonhomogeneous Poisson Process (NHPP) . 1
47、08 4.7.3 Impact of Assumption Regarding Repair 116 4.8 Treatment of Uncertain Data 118 4.8.1 Uncertainty in Binomial Demands or Poisson Exposure Time 118 4.8.2 Uncertainty in Binomial or Poisson Failure Counts . 121 4.8.3 Censored Data for Random Durations 132 4.8.4 Legacy and Heritage Data . 136 4.9 Bayesian Regression Models 144 4.10 Using Higher-Level Data to Estimate Lower-Level Parameters . 147 4.11 Using Information Elicited from Experts 150 4.11.1 Using Information from a Single Expert . 150 4.11.2 Using Information from Multiple Experts . 151 4.11.3 Bayesian Inference
