RBI and Bayes' rule
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The document discusses the application of Bayes' theorem in Risk-Based Inspection (RBI) to update prior knowledge of degradation rates through inspection data. It outlines the historical significance of Bayesian methods in various fields, including their use in damage state classification and inspection effectiveness evaluation. The paper emphasizes the importance of using rigorous statistical methods to improve the accuracy of damage detection and assessment in equipment maintenance.
RBI and Bayes' rule
- 1. 1 Worked Example RBI and Bayes’ theorem Draft for discussion only “... we balance probabilities and choose the most likely. It is the scientific use of the imagination ... ” Sherlock Holmes, The Hound of the Baskervilles. AC Doyle, 1901. copyright © riccardo.cozza@gmail.com
- 2. 2 Bayes’ rule is a rigorous method for interpreting evidence in the context of previous experience or knowledge. It was discovered by Thomas Bayes (c. 1701-1761), and independently discovered by Pierre-Simon Laplace (1749- 1827). After more than two centuries of controversy, during which Bayesian methods have been both praised and pilloried, Bayes’ rule has recently emerged as a powerful tool with a wide range of applications, which include: genetics , linguistics , image processing, brain imaging, cosmology, machine learning, epidemiology, psychology, forensic science, human object recognition, evolution , visual perception, ecology and even the work of the fictional detective Sherlock Holmes . Historically, Bayesian methods were applied by Alan Turing to the problem of decoding the German enigma code in the Second World War, but this remained secret until recently. In essence, Bayes’ rule provides a method for not fooling ourselves into believing our own prejudices, because it represents a rational basis for believing things that are probably true, and for disbelieving things that are probably not. Introduction (1) Draft for discussion only Note (1): “Bayes’ Rule - A Tutorial Introduction to Bayesian Analysis –” James V Stone
- 3. 3 RBI uses Bayes’ theorem to update the prior knowledge of the degradation rate with the information gained from an inspection. RBI and Bayes’ theorem Inspection Data Posteriori Probability Since it can be very difficult to find a representative continuous prior density for the degradation rate such that the posterior is easily calculated, RBI uses a discrete version of Bayes’ theorem. Damage Detection Probability Bayes theorem Prior Probability Draft for discussion only
- 4. 4 In RBI three generic damage states are defined. They are noted 1, 2 and 3 in increasing gravity order that are a priori susceptible to occur. RBI and Bayes’ theorem Damage State Damage State Range of actual damage state 1 Predicted "rate" or less 2 Predicted "rate" to two times "rate" 3 Two to four times predicted "rate" Draft for discussion only
- 5. 5 RBI and Bayes’ theorem Inspection Effectiveness Inspection Category Qualitative Inspection Effectivness Category Description A Highly Effective Highly Effective The inspection methods will correctly identify the true damage state in nearly every case (or 80–100% confidence). B Usually Effective Usually Effective The inspection methods will correctly identify the true damage state most of the time (or 60–80% confidence). C Fairly Effective Fairly Effective The inspection methods will correctly identify the true damage state about half of the time (or 40–60% confidence). D Poorly Effective Poorly Effective The inspection methods will provide little information to correctly identify the true damage state (or 20–40% confidence). E Ineffective The inspection method will provide no or almost no information that will correctly identify the true damage state and are considered ineffective for detecting the specific damage mechanism (less then 20% confidence) The probability of detection of a damage involves directly the inspection effectiveness notion. This effectiveness evaluation implements results obtained from several statistical analysis developed by organisms like Nordtest Europe, EPRI Nippon steel or US Navy. Those tests, aiming to determine the probability of detection (POD) of different non destructive tests, are conducted using tests on standard block. More precisely, those tests allow to determine a given inspection effectiveness probability to reveal the equipment actual damage state. Then, detection effectiveness, which depends on the appropriateness of the selected inspection method with the searched potential damage mechanism, is linked with the three generic damage states. The detection effectiveness is the probability to actually detect the expected state. Draft for discussion only
- 6. 6 RBI and Bayes’ theorem Inspection Effectiveness 1,0 0,8 0,6 0,4 0,2 0,0 543210 Density Damage rate factor 0,03 0,15 0,80 1,0 0,8 0,6 0,4 0,2 0,0 543210 Density Damage rate factor 0,05 0,20 0,70 1,0 0,8 0,6 0,4 0,2 0,0 543210 Density Damage rate factor 0,10 0,30 0,50 Damage State Range of actual damage state Likelihood that inspection result determines the true damage state Poorly or Ineffective Fairly Effective Usually Effective Higly Effective 1 Predicted "rate" or less 0.33 0.5 0.7 0.8 2 Predicted "rate" to two times "rate" 0.33 0.3 0.2 0.15 3 Two to four times predicted "rate" 0.33 0.2 0.1 0.05 0,6 0,4 1,0 0,8 0,0 0,2 543210 Damage rate factor Density 0,33 0,33 0,17 Poorly/Ineffective Fairly Effective Usually Effective Higly Effective Draft for discussion only
- 7. 7 RBI uses Bayes’ theorem to update the prior knowledge of the degradation rate with the information gained from an inspection. Since it can be very difficult to find a representative continuous prior density for the degradation rate such that the posterior is easily calculated, RBI uses a discrete version of Bayes’ theorem: RBI and Bayes’ theorem Bayes’ theorem ( ) )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 313k212k111k 1k i rrrrrrrrr rrr kkk iki k −−− − ++ = πππ π π where: • i = 1, 2, 3 ≡ damage state; • k = 1,…,m ≡ time step; • m = number of time steps considered in the planning horizon; • r = observed corrosion rate estimated from inspection; • r1 = r , r1 = 2*r , r1 = 4*r = corrosion rates related to damage states 1, 2 and 3, respectively; • πk-1(ri) = prior probability of the damage state i at time k; • Lk(r|ri) = likelihood of observing the result r of an inspection performed at k given that the equipment item is under the damage state i; • πk(r|ri) ≡ posterior distribution for the damage state k. Note that πk(r|ri) becomes the prior distribution when the next inspection takes place, which permits the Bayesian updating of the degree of confidence on r. The likelihood Lk(r|ri) depends on the effectiveness of the inspection technique. Indeed, the Table on pag. 5 quantitatively expresses this classification as the likelihood that the observed damage state (collected from an inspection program) actually represents the true state. Thus, the above equation, provides a manner to update the degree of confidence based on the inspection effectiveness. In this way, it is expected that the knowledge acquired from the inspection program reduces the uncertainty about the actual deterioration state of the equipment Draft for discussion only
- 8. 8 InformationSourcesfor DamageRate Low Confidence Data Moderate Confidence Data High Confidence Data • Published data. • Corrosion rate tables. • “Default” values. Although they are often used for design decisions, the actual corrosion rate that will be observed in a given process situation may significantly differ from the design value. Laboratory testing with simulated process conditions. Limited in-situ corrosion coupon testing. Corrosion rate data developed from sources that simulate the actual process conditions usually provide a higher level of confidence in the predicted corrosion rate. Extensive field data from thorough inspections. Coupon data, reflecting five or more years of experience with the process equipment (assuming no change in process conditions has occurred). If enough data are available from actual process experience, there is little likelihood that the actual corrosion rate will greatly exceed the expected value under normal operating conditions. We start with a first estimation of the equipment state, according to the available data reliability. For example, we have High Confidence Data. RBI and Bayes’ theorem Prior distribution Draft for discussion only
- 9. 9 We start with a first estimation of the equipment state, according to the available data (a priori estimation). For example, we have High Confidence Data. A priori Damage Rate Range Confidence in Predicted Damage Rate Low Confidence Data Moderate Confidence Data High Confidence Data Predicted "rate" or less 0.5 0.7 0.80 Predicted "rate" to two times "rate" 0.3 0.2 0.15 Two to four times predicted "rate" 0.2 0.1 0.05 RBI and Bayes’ theorem Inspection updating 1,0 0,8 0,6 0,4 0,2 0,0 543210 Density Damage rate factor 0,05 0,20 0,70 1,0 0,8 0,6 0,4 0,2 0,0 543210 Density Damage rate factor 0,10 0,30 0,50 Low Confidence Data Moderate Confidence Data 0,6 1,0 0,8 0,4 0,2 0,0 543210 Density 0,15 0,80 Damage rate factor 0,03 High Confidence Data Draft for discussion only
- 10. 10 For example, we have “Usually effective” inspection, The first inspection has confirmed the expectation damage rate (state 1). RBI and Bayes’ theorem Inspection updating Lk(r|ri) Conditional Probability of Inspection “E” None or Ineffective “D” Poorly Effective “C” Fairly Effective “B” Usually Effective “A” Highly Effective Predicted "rate" or less 0.33 0.4 0.5 0.7 0.9 Predicted "rate" to two times "rate" 0.33 0.33 0.3 0.2 0.09 Two to four times predicted "rate" 0.33 0.27 0.2 0.1 0.01 In these conditions, before inspection, we have the following probabilities: Damage State Prior probability π0(ri) Damage detection probability L1(r|ri) 1 0.80 0.7 2 0.15 0.2 3 0.05 0.1 0,8 3210 0,4 0,2 0,0 54 1,0 0,6 0,05 0,20 Damage rate factor 0,70 Density Usually Effective Draft for discussion only
- 11. 11 After the inspection, confirming the expectation damage rate (State 1), using Bayes theorem: RBI and Bayes’ theorem Posteriori distribution ( ) ( ) ( ) 0.008 )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 0.050 )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 0.941 0.05*1.015.0*2.00.80*0.7 0.80*0.7 )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 303120211011 3031 31 303120211011 2121 21 303120211011 1011 11 = ++ = = ++ = = ++ = ++ = − rrrrrrrrr rrr rrrrrrrrr rrr rrrrrrrrr rrr k πππ π π πππ π π πππ π π 0,5 0,4 0,3 0,2 0,1 0,0 1,0 0,9 0,8 0,7 0,6 Damage status -84% -67% Probability +18% 3 0,01 0,05 2 0,05 0,15 1 0,94 0,80 Posteriori evaluation (Probability Bayes Update) A priori evaluation Draft for discussion only
- 12. 12 RBI and Bayes’ theorem Posteriori distribution 0,0 1,0 1,5 0,5 Probability 0,234 Not 3 -51%+245% -1% 0,475 Not 2 0,4190,425 Not 1 0,345 0,100 A priori evaluation Probability Bayes Update If, on the other hand, the inspection has not confirmed the expectation damage rate, using Bayes theorem : Damage State State Corrosion rate probability Damage detection probability Π0(ri) Π0(r̅i) L1(r|ri) =L1(r̅|r̅i) Not 1 0.80 (1-0.80)/((1-0.8)+(1-0.15)+(1-0.05)) = 0.100 0.7 Not 2 0.15 (1-0.15)/((1-0.8)+(1-0.15)+(1-0.05)) = 0.425 0.2 Not 3 0.05 (1-0.05)/((1-0.8)+(1-0.15)+(1-0.05)) = 0.475 0.1 ( ) )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 313k212k111k 1k i rrrrrrrrr rrr kkk iki k −−− − ++ = πππ π π ( ) ( ) ( ) 234.0 )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 419.0 )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 345.0 475.0*1.0425.0*2.01.0*7.0 1.0*7.0 )(*)|(L)(*)|(L)(*)|(L )(*)|(L rr 303120211011 3031 31 303120211011 2021 21 303120211011 1011 11 = ++ = = ++ = = ++ = ++ = rrrrrrrrr rrr rrrrrrrrr rrr rrrrrrrrr rrr πππ π π πππ π π πππ π π 0,0 1,0 1,5 0,5 Probability 0,234 3 +368% -57% +180% 0,050 2 0,420 0,150 1 0,345 0,800 A priori evaluation Probability Bayes Update Draft for discussion only ( ) ( )r|rπrrπ ikik =
- 13. 13 Draft for discussion only RBI and Bayes’ theorem Posteriori distribution: confirming vs. not confirming damage state 0,4 0,2 0,0 1,0 0,8 0,6 Damage status Probability 3 0,01 2 0,05 1 0,94 Posteriori evaluation 0,4 0,2 0,0 1,0 0,8 0,6 Damage status Probability 3 0,23 2 0,42 1 0,35 Posteriori evaluation 0,4 0,2 0,0 1,0 0,8 0,6 Damage status Probability 3 0,05 2 0,15 1 0,80 A priori evaluation the inspection has not confirmed the expectation damage rate the inspection has confirmed the expectation damage rate
- 14. 14 Back up copyright © riccardo.cozza@gmail.com
- 15. 15 6,1% 20,4% 3 8,4% 23,1% 68,5% 2 11,5% 25,7% 62,9% 1 15,3% 73,5% Inspection number 100% 6 3,1%15,4% 28,0% 56,7% A priori 20,0% 30,0% 50,0% 81,5% 5 4,4% 17,8% 77,8% 4 0,4% 4,4% 95,2% 4 7,1% 3 5,0% 16,9% 78,1% 1 10,5% 91,9% 50,0% 97,1% 65,8% Inspection number 2,7% 2 0,2% 86,6% 5 11,2% A priori 6 20,0% 100% 2,2% 0,9% 30,0% 23,7% Damage state 3 Damage state 2 Damage state 1 100% 6 0,00% 0,03% 99,97% 0,00%0,02% 5 99,88% Inspection number 0,11% 4 0,40% 99,59% 3 0,11% 1,38% 98,51% 13,95% 81,40% A priori 20,00% 30,00% 50,00% 2 0,77% 4,63% 94,59% 1 4,65% RBI and Bayes’ theorem Confidence in predicted Damage Rate: Low Reliability Data 100% 0,00% 100,00% 4 0,00% 0,01% 99,99% 3 0,00% 0,06% 99,94% 2 0,00% 0,60% 99,40% A priori 0,00% 0,00% 100,00% 50,00% 93,95% Inspection number 6 5,64% 30,00% 0,42% 0,00% 1 20,00% 5 Inspection Effectiveness: POORLY Inspection Effectiveness: FARLY Draft for discussion only Inspection Effectiveness: USUALLY Inspection Effectiveness: HIGHLY
- 16. 16 Probability of Detection (2) Draft for discussion only Note (2): “Fundamentals of Structural Integrity” Alten F. Grandt Jr. N total specimens N=N1+N2 N1 specimens containing cracks N2 specimens are un-cracked A1 cracks are detected (correct rejection) A2 cracks are undetected (incorrect acceptance) A3 cracks are detected (false rejection) A4 no cracks are detected (correct acceptance) Type II error (safety problem) Type I error (economic problem) Consider the potential results of an inspection of N specimens. N1 specimens contain A1 + A2 cracks, while N2 specimens are uncracked (N = N1 + N2). Four possible results of the N1 cracked and N2 uncracked specimens can be used to define the following quantities: • Probability of detection (POD)= sensitivity of detection = A1/N1 • Probability of recognition (POR) = A4/ N2 • False-call probability (FCP) = A3/ N2 • Accuracy of the observer = (A1 + A4)/N