Worked Example
RBI and Bayes’ theorem
“... 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
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Introduction (1)
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 (17491827).
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.
Note (1): “Bayes’ Rule - A Tutorial Introduction to Bayesian Analysis –” James V Stone
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RBI and Bayes’ theorem
RBI uses Bayes’ theorem to update the prior knowledge of the degradation rate with the information gained from an
inspection.
Prior
Probability
Damage Detection
Probability
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.
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RBI and Bayes’ theorem
Damage State
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.
Damage State
Range of actual damage state
The damage in the equipment is “no” worse than what
1 is expected based on damage rate models or Predicted "rate" or less
experience.
The damage in the equipment is “somewhat” worse
2 than anticipated. This level of damage is sometimes Predicted "rate" to two times "rate"
seen in similar equipment items.
The damage in the equipment is “considerably” worse
than anticipated. This level of damage is rarely seen in
3
Two to four times predicted "rate"
similar equipment items, but has been observed on
occasion industry wide.
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RBI and Bayes’ theorem
Inspection Effectiveness
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.
Inspection Qualitative Inspection
Category Effectivness Category
A
Highly Effective
B
Usually Effective
C
Fairly Effective
D
Poorly Effective
E
Ineffective
Description
Highly Effective The inspection methods will correctly identify the true damage state in nearly every
case (or 80–100% confidence).
Usually Effective The inspection methods will correctly identify the true damage state most of the
time (or 60–80% confidence).
Fairly Effective The inspection methods will correctly identify the true damage state about half of the
time (or 40–60% confidence).
Poorly Effective The inspection methods will provide little information to correctly identify the true
damage state (or 20–40% confidence).
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)
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RBI and Bayes’ theorem
Inspection Effectiveness
1
2
3
Range of actual damage state
Predicted "rate" or less
Predicted "rate" to two times "rate"
Two to four times predicted "rate"
Poorly/Ineffective
Fairly Effective
0,6
0,4
0,6
0,50
0,4
0,30
0,2
0,33 0,33
0,10
1
2
3
4
Damage rate factor
5
0,70
0,6
0,4
1
2
3
Damage rate factor
4
5
0,6
0,4
0,20
0,15
0,2
0,03
0,0
0
Higly Effective
0,80
0,8
0,05
0,0
0
Usually Effective
0,2
0,17
0,0
1,0
0,8
Density
0,8
Density
Density
0,8
0,2
1,0
1,0
1,0
Density
Damage
State
Likelihood that inspection result determines the true
damage state
Poorly or
Fairly
Usually
Higly
Ineffective
Effective
Effective
Effective
0.33
0.5
0.7
0.8
0.33
0.3
0.2
0.15
0.33
0.2
0.1
0.05
0,0
0
1
2
3
Damage rate factor
4
5
0
1
2
3
4
5
Damage rate factor
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RBI and Bayes’ theorem
Bayes’ theorem
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:
 k (ri r ) =
where:
•
•
•
•
•
•
•
L k (r | ri ) *  k −1 (ri )
L k (r | r1 ) *  k −1 (r1 ) + L k (r | r2 ) *  k −1 (r2 ) + L k (r | r3 ) *  k −1 (r3 )
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
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RBI and Bayes’ theorem
Prior distribution
Information Sources for
Damage Rate
We start with a first estimation of the equipment state, according to the available data reliability.
For example, we have High Confidence Data.
Low
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.
Moderate
Confidence Data
High
Confidence Data
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.
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RBI and Bayes’ theorem
Inspection updating
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
Low
Confidence Data
Confidence in Predicted Damage Rate
Moderate
Confidence Data
High
Confidence Data
0.5
0.7
0.80
0.3
0.2
0.15
0.2
0.1
0.05
Predicted "rate"
or less
Predicted "rate"
to two times
"rate"
Two to four
times predicted
"rate"
1,0
1,0
Moderate Confidence Data
Low Confidence Data
0,8
0,50
0,4
0,30
0,2
0,6
0,4
1
2
3
Damage rate factor
4
5
0,4
0,15
0,03
0,0
0
0,6
0,2
0,05
0,0
High Confidence Data
0,20
0,2
0,10
0,80
0,8
0,70
Density
0,6
Density
Density
0,8
1,0
0,0
0
1
2
3
Damage rate factor
4
5
0
1
2
3
4
5
Damage rate factor
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RBI and Bayes’ theorem
Inspection updating
For example, we have “Usually effective” inspection,
The first inspection has confirmed the expectation damage rate (state 1).
Conditional
Probability of
Inspection
Predicted
"rate" or less
Predicted
"rate" to two
times "rate"
Two to four
times predicted
"rate"
Lk(r|ri)
“E” None or
Ineffective
“D” Poorly
Effective
“C” Fairly
Effective
“B” Usually
Effective
“A” Highly
Effective
0.33
0.4
0.5
0.7
0.9
0.33
0.33
0.3
0.2
0.09
0.33
0.27
0.2
0.1
0.01
In these conditions, before inspection, we have the following probabilities:
Prior probability
π0(ri)
1
2
3
0.80
0.15
0.05
Damage detection
probability
L1(r|ri)
0.7
0.2
0.1
Usually Effective
0,8
Density
Damage
State
1,0
0,70
0,6
0,4
0,20
0,2
0,05
0,0
0
1
2
3
Damage rate factor
4
5
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RBI and Bayes’ theorem
Posteriori distribution
After the inspection, confirming the expectation damage rate (State 1), using Bayes theorem:
 1 (r1 r ) =
L1 (r | r1 ) *  0 (r1 )
0.7 * 0.80
=
= 0.941
L1 (r | r1 ) *  0 (r1 ) + L1 (r | r2 ) *  0 (r2 ) + L1 (r | r3 ) *  0 (r3 ) 0.7 * 0.80 + 0.2 * 0.15 + 0.1* 0.05
 1 (r2 r ) =
L1 (r | r2 ) *  k −1 (r2 )
= 0.050
L1 (r | r1 ) *  0 (r1 ) + L1 (r | r2 ) *  0 (r2 ) + L1 (r | r3 ) *  0 (r3 )
 1 (r3 r ) =
L1 (r | r3 ) *  0 (r3 )
= 0.008
L1 +18%
(r | r1 ) *  0 (r1 ) + L1 (r | r2 ) *  0 (r2 ) + L1 (r | r3 ) *  0 (r3 )
1,0
0,94
A priori evaluation
0,9
0,8
Posteriori evaluation (Probability Bayes Update)
0,80
Probability
0,7
0,6
0,5
0,4
0,3
-67%
0,2
-84%
0,15
0,1
0,0
0,05
1
2
0,05
0,01
3
Damage status
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RBI and Bayes’ theorem
Posteriori distribution
If, on the other hand, the inspection has not confirmed the expectation damage rate, using Bayes theorem :
( )
 k ri r =
L k (r | ri ) *  k −1 (ri )
L k (r | r1 ) *  k −1 (r1 ) + L k (r | r2 ) *  k −1 (r2 ) + L k (r | r3 ) *  k −1 (r3 )
Damage
State
Π0(ri)
0.80
0.15
0.05
Not 1
Not 2
Not 3
Π0(r̅i)
(1-0.80)/((1-0.8)+(1-0.15)+(1-0.05)) = 0.100
(1-0.15)/((1-0.8)+(1-0.15)+(1-0.05)) = 0.425
(1-0.05)/((1-0.8)+(1-0.15)+(1-0.05)) = 0.475
( )
L1 (r | r1 ) *  0 (r1 )
0.7 * 0.1
=
= 0.345
L1 (r | r1 ) *  0 (r1 ) + L1 (r | r2 ) *  0 (r2 ) + L1 (r | r3 ) *  0 (r3 ) 0.7 * 0.1 + 0.2 * 0.425 + 0.1* 0.475
( )
L1 (r | r2 ) *  0 (r2 )
= 0.419
L1 (r | r1 ) *  0 (r1 ) + L1 (r | r2 ) *  0 (r2 ) + L1 (r | r3 ) *  0 (r3 )
( )
L1 (r | r3 ) *  0 (r3 )
= 0.234
L1 (r | r1 ) *  0 (r1 ) + L1 (r | r2 ) *  0 (r2 ) + L1 (r | r3 ) *  0 (r3 )
 1 r1 r =
 1 r2 r =
 1 r3 r =
-1%
+245%
0,5
0,345
0,425 0,419
Not 1
A priori evaluation
0,475
0,234
0,100
0,0
-51%
Not 2
Not 3
Probability Bayes Update
( )
π k ri r = π k (ri | r )
1,5
-57%
Probability
Probability
1,5
1,0
Damage detection
probability
L1(r|ri) =L1(r̅|r̅i)
0.7
0.2
0.1
State Corrosion rate probability
1,0
0,5
+180%
0,800
+368%
0,420
0,345
0,150
0,234
0,050
0,0
1
A priori evaluation
2
3
Probability Bayes Update 12
RBI and Bayes’ theorem
Posteriori distribution: confirming vs. not confirming damage state
1,0
Posteriori evaluation
the inspection has confirmed
the expectation damage rate
Probability
0,8
1,0
0,8
0,94
0,6
0,4
0,2
0,05
A priori evaluation
0,80
1
0,6
2
3
Damage status
0,4
1,0
0,15
0,2
Posteriori evaluation
0,05
0,8
0,0
1
2
3
Damage status
the inspection has not
confirmed the expectation
damage rate
Probability
Probability
0,01
0,0
0,6
0,42
0,4
0,35
0,23
0,2
0,0
1
2
3
Damage status
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Back up
copyright © riccardo.cozza@gmail.com
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RBI and Bayes’ theorem
Confidence in predicted Damage Rate: Low Reliability Data
62,9% 68,5% 73,5% 77,8%
25,7% 23,1% 20,4% 17,8%
11,5% 8,4% 6,1% 4,4%
100%
65,8% 78,1% 86,6% 91,9% 95,2%
23,7% 16,9% 11,2% 7,1% 4,4%
10,5% 5,0% 2,2% 0,9% 0,4%
100%
50,0%
50,0% 56,7%
81,5%
30,0%
30,0%
28,0%
15,4%3,1%
Inspection number
6
20,0% 15,3%
A priori
97,1%
1
2
3
4
5
20,0%
A priori
Inspection Effectiveness: POORLY
81,40% 94,59% 98,51% 99,59%
13,95% 4,63% 1,38% 0,40%
4,65% 0,77% 0,11% 0,02%
1
2
3
4
5
6
Inspection Effectiveness: FARLY
99,88%
0,11%
0,00%
100%
50,00%
93,95% 99,40% 99,94% 99,99%
5,64% 0,60% 0,06% 0,01%
0,42% 0,00% 0,00% 0,00%
100,00%
0,00%
0,00%
100%
50,00%
99,97%
30,00%
100,00%
30,00%
20,00%
A priori
2,7%
0,2%
Inspection number
1
2
3
4
5
Inspection Effectiveness: USUALLY
6
0,03%
0,00%
Inspection number
20,00%
A priori
1
2
3
4
5
6
0,00%
0,00%
Inspection number
Inspection Effectiveness: HIGHLY
Damage state 1
Damage state 2
Damage state 3
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Probability of Detection (2)
Consider the potential results of an inspection of N specimens.
N total
specimens
N=N1+N2
N1 specimens
containing cracks
A1 cracks are
detected
(correct rejection)
N2 specimens are
un-cracked
A2 cracks are
undetected
(incorrect acceptance)
A3 cracks are
detected
(false rejection)
Type II error
(safety problem)
Type I error
(economic problem)
A4 no cracks are
detected
(correct acceptance)
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
Note (2): “Fundamentals of Structural Integrity” Alten F. Grandt Jr.
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