Bayesian Network Approach To Assessing System Reliability .
Bayesian Network Approach to Assessing System Reliability for Improving SystemDesign and Optimizing System MaintenancebyDongjin LeeA Dissertation Presented in Partial Fulfillmentof the Requirements for the DegreeDoctor of PhilosophyApproved March 2018 by theGraduate Supervisory Committee:Rong Pan, ChairDouglas C. MontgomeryTeresa WuXiaoping DuARIZONA STATE UNIVERSITYMay 2018
ABSTRACTA quantitative analysis of a system that has a complex reliability structure alwaysinvolves considerable challenges. This dissertation mainly addresses uncertainty inherent in complicated reliability structures that may cause unexpected and undesiredresults.The reliability structure uncertainty cannot be handled by the traditional reliability analysis tools such as Fault Tree and Reliability Block Diagram due to theirdeterministic Boolean logic. Therefore, I employ Bayesian network that provides aflexible modeling method for building a multivariate distribution. By representing asystem reliability structure as a joint distribution, the uncertainty and correlationsexisting between system’s elements can effectively be modeled in a probabilistic manner. This dissertation focuses on analyzing system reliability for the entire systemlife cycle, particularly, production stage and early design stages.In production stage, the research investigates a system that is continuously monitored by on-board sensors. With modeling the complex reliability structure byBayesian network integrated with various stochastic processes, I propose severalmethodologies that evaluate system reliability on real-time basis and optimize maintenance schedules.In early design stages, the research aims to predict system reliability based onthe current system design and to improve the design if necessary. The three mainchallenges in this research are: 1) the lack of field failure data, 2) the complex reliability structure and 3) how to effectively improve the design. To tackle the difficulties,I present several modeling approaches using Bayesian inference and nonparametricBayesian network where the system is explicitly analyzed through the sensitivityanalysis. In addition, this modeling approach is enhanced by incorporating a temporal dimension. However, the nonparametric Bayesian network approach generallyi
accompanies with high computational efforts, especially, when a complex and largesystem is modeled. To alleviate this computational burden, I also suggest to buildinga surrogate model with quantile regression.In summary, this dissertation studies and explores the use of Bayesian networkin analyzing complex systems. All proposed methodologies are demonstrated by casestudies.ii
ACKNOWLEDGMENTSFirst of all, I must show my sincere gratitude to Dr. Rong Pan for mentoring mewith generosity, patient, and financial support. His knowledge and encouragementhave guided me into reliability engineering and made my graduate school experienceas a great one. I also thank to my committee members Dr. Douglas Montgomery,Dr. Teresa Wu, and Dr. Xiaoping Du. The key ingredient of writing this dissertationis their support.I would like thank to my parents and parents in law. Their constant support andencouragement are the most valuable source for me to go through the challenginggraduate school life.I cannot find adequate words to thank my wife, Yeawon Yoo. I deeply appreciateher dedication despite she is also having tough time in the same Ph.D. program.iii
TABLE OF CONTENTSPageLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixCHAPTER1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Dissertation Overview and Organization . . . . . . . . . . . . . . . . . . . . . . . . . .22 PREDICTIVE MAINTENANCE OF COMPLEX SYSTEMS WITH MULTILEVEL RELIABILITY STRUCTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.2Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.3Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.42.52.3.1Markov Chain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.3Proposed Methods for Predictive Maintenance . . . . . . . . . . . . . 14Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.1Estimating Transition Matrices and Goodness of Fit . . . . . . . . 202.4.2Implementing PdM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 A NONPARAMETRIC BAYESIAN NETWORK APPROACH TO ASSESSING SYSTEM RELIABILITY AT EARLY DESIGN STAGES . . . . . 313.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3Early Phases of Engineering Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1Conceptual Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35iv
CHAPTER3.4Page3.3.2Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.3Embodiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Non-parametric Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.1NPBNs and Vines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.2Eliciting Expert Opinions and Bayesian Inference . . . . . . . . . . 433.5Application of NPBN to System Reliability Assessment . . . . . . . . . . . . 463.6Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.73.6.1Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.6.2Simulation Result and Sensitivity Analysis . . . . . . . . . . . . . . . . . 54Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 SYSTEM RELIABILITY DESIGN VIA A NONPARAMETRIC BAYESIANNETWORK APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.1Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.2Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3Non-parametric Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.44.54.3.1NPBN and D-vine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.2Sampling D-vines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Estimation of Conditional Rank Correlation . . . . . . . . . . . . . . . . . . . . . . 734.4.1Elicitation of Expert Opinions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4.2Bayesian Information Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Modeling Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.5.1Node Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77v
CHAPTERPage4.5.2Estimation of Parameters for Continuous-Time Systems . . . . 784.6Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.7Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 RELIABILITY-BASED DESIGN OPTIMIZATION FOR OPTIMAL SYSTEM RELIABILITY USING NONPARAMETRIC BAYESIAN NETWORK AND QUANTILE REGRESSION SURROGATE . . . . . . . . . . . . . . 905.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2Reliability Analysis with NPBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.35.2.1NPBN Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2.2System Reliability Analysis with NPBN . . . . . . . . . . . . . . . . . . . 99Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3.1Quantile Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3.2Bayesian Quantile Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.6Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.7Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.1Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122APPENDIXA FURTHER MATERIALS FOR CHAPTER 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 131B FURTHER MATERIALS FOR CHAPTER 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 135C FURTHER MATERIALS FOR CHAPTER 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 137vi
LIST OF TABLESTablePage2.1Marginal Probability Tables for the Root Nodes . . . . . . . . . . . . . . . . . . . . . . 132.2Conditional Probability Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3Examples of Backward Recurrence Time and Semi-markov Chain . . . . . . 192.4Laser Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5Fatigue Crack Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6Discretization Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7Fit Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.8System Reliability Forecast When the Current Component State Is(0,0,0,0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.9System Reliability Forecast When the Current Component State Is(1,0,1,0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.10 System Reliability Forecast When the Current Component State Is(1,0,1,2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1Expected Components’ Performance and System Reliability at Time t . 553.2Results of the Morris Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.1Degradation Data of Two Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2Increments of Two Degradation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3Degradation Models and Lifetime Distributions for Components . . . . . . . 814.4Failure Probability of Each Component at Time 40, 42, 44, and 46 . . . . 814.5Degradation Models and Lifetime Distributions for Components . . . . . . . 834.6Failure Probabilities of Three Components Calculated from Their Degradation Models at Time Points 45, 45.7 and 46 . . . . . . . . . . . . . . . . . . . . . . . . 854.7Parameters of the Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.8Reliability Allocation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87vii
TablePage5.1Prior Distributions for Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2A Summary of the Final Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3RBDO Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.1 The CPT for Subsystem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132A.2 The CPT for Subsystem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.3 The CPT for System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134viii
LIST OF FIGURESFigurePage2.1A BN Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2Multi-Level Hierarchical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3Laser Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4Fatigue Crack Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5Reliability Structure of the Hypothetical System . . . . . . . . . . . . . . . . . . . . . 242.6System Reliability Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7System Reliability Forecast with Realizations . . . . . . . . . . . . . . . . . . . . . . . . 273.1An Example of NPBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2Parameterized NPBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3D-vine on Three Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4D-vine Represented by Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5A D-vine on n Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6NPBN on X1 , X2 , X3 , and X4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7D-vine on X3 , X1 , and X2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.8D-vine on x3 , X1 , and x2 by Paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.9D-vine on x4 , X1 , X3 , and x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.10 D-vine on x4 , X1 , X3 , and x2 by Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.11 System’s Performance Where x2 .95 and x3 .9 . . . . . . . . . . . . . . . . . . . 503.12 System’s Performance Where x2 .1 and x3 .3 . . . . . . . . . . . . . . . . . . . . . 503.13 Component 2’s Performance Where x1 .9 . . . . . . . . . . . . . . . . . . . . . . . . . . 503.14 Component 3’s Performance Where x1 .9 . . . . . . . . . . . . . . . . . . . . . . . . . . 503.15 Component 2’s Performance Where x1 .1 . . . . . . . . . . . . . . . . . . . . . . . . . . 513.16 Component 3’s Performance Where x1 .1 . . . . . . . . . . . . . . . . . . . . . . . . . . 513.17 Component 3’s Performance Where x1 .1 and x2 .5 . . . . . . . . . . . . . . . 51ix
FigurePage3.18 Component 3’s Performance Where x1 .9 and x2 .5 . . . . . . . . . . . . . . . 513.19 Lower Tail Dependence of Clayton Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.20 Upper Tail Dependence of Gumbel Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.21 Fault Tree of the Lithium-Chip Manufacturing Process . . . . . . . . . . . . . . . 533.22 NPBN Structure of the Lithium-Chip Manufacturing Process . . . . . . . . . 543.23 Parameterized NPBN of the Lithium-Chip Manufacturing Process . . . . . 543.24 Component 1’s Expected Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.25 Component 2’s Expected Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.26 Component 6’s Expected Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.27 Component 7’s Expected Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1Failure Probability of Component over Time . . . . . . . . . . . . . . . . . . . . . . . . . 634.2Framework of the Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3A Simple NPBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4D-vine on Two Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.5D-vine on Three Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.6An Example of Gumbel Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.7The Parameterized NPBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.8Simulation at Time 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.9Simulation at Time 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.10 Simulation at Time 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.11 Simulation at Time 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.12 A System Consisting of a Subsystem and Components . . . . . . . . . . . . . . . . 834.13 Parameterized NPBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.14 Monte Carlo Simulation Results at Time 45, 45.7 and 46 . . . . . . . . . . . . . . 85x
FigurePage4.15 A Pareto Frontier of Alternative Design Solutions . . . . . . . . . . . . . . . . . . . . 885.1A BN Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2A Distribution of x1 given Evidence of x2 and x3 . . . . . . . . . . . . . . . . . . . . . 975.3A D-vine Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.4Skewed Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.5An Example of LR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.6Examples of LR and QRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.7A Simulation Result of System’s FP with Design 1 . . . . . . . . . . . . . . . . . . . 1065.8A Simulation Result of System’s FP with Design 2 . . . . . . . . . . . . . . . . . . . 1065.9The Reliability Structure of Radar System . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.10 The Distribution of Predicted System Performance Based on the Current Design . . . . . . . . . . . . . . . .
posing system reliability into component reliability in a deterministic manner (i.e., series or parallel systems). Consequentially, any popular reliability analysis tools such as Fault Tree and Reliability Block Diagram are inadequate. In order to overcome the challenge, this dissertation focuses on modeling system reliability structure using
Computational Bayesian Statistics An Introduction M. Antónia Amaral Turkman Carlos Daniel Paulino Peter Müller. Contents Preface to the English Version viii Preface ix 1 Bayesian Inference 1 1.1 The Classical Paradigm 2 1.2 The Bayesian Paradigm 5 1.3 Bayesian Inference 8 1.3.1 Parametric Inference 8
value of the parameter remains uncertain given a nite number of observations, and Bayesian statistics uses the posterior distribution to express this uncertainty. A nonparametric Bayesian model is a Bayesian model whose parameter space has in nite dimension. To de ne a nonparametric Bayesian model, we have
Bayesian methods, we provide evidence that Bayesian interval estimators perform well compared to available frequentist estimators, under frequentist performance criteria. The Bayesian non-parametric approach attempts to uncover and exploit structure in the data. For example, if the e
Bayesian methods are inherently small sample, they are a coherent choice. Even in the absence of a direct motivation for using Bayesian methods, we provide evidence that Bayesian interval estimators perform well compared to available freque
Key words Bayesian networks, water quality modeling, watershed decision support INTRODUCTION Bayesian networks A Bayesian network (BN) is a directed acyclic graph that graphically shows the causal structure of variables in a problem, and uses conditional probability distributions to define relationships between variables (see Pearl 1988, 1999;
Alessandro Panella (CS Dept. - UIC) Probabilistic Representation and Reasoning May 4, 2010 14 / 21. Bayesian Networks Bayesian Networks Bayesian Networks A Bayesian (or belief) Network (BN) is a direct acyclic graph where: nodes P i are r.v.s
csv file with individual data for Bayesian network structure learning and parameter training. The data is an N M matrix with discrete data, where N is the number of observables and M is the number of the features (nodes). Network construction Bayesian network constructions are performed using the methods in the bnlearn R package [6]. Users can .
Intro — Introduction to Bayesian analysis . Bayesian analysis is a statistical analysis that answers research questions about unknown parameters of statistical models by using probability statements. Bayesian analysis rests on the assumption that all . Proportion infected in the population, q p(q) p(q y)
