ON THE USE/APPLICABILITY OF
BAYES’ THEOREM
TO
FOOD/DRUG SAFETY
by Stan Kaplan
Presented to
1st International Conference on Microbiological Risk Assessment
July 24-26, 2002
Standard Derivation
of Bayes’ Theorem
Probability
=
of A and B
P(A^B)
= P(A)
}
P(B/A)
= P(B) P(A/B)
Axiom
of
Probability theory
Therefore: P(A)P(B/A) = P(B) P(A/B)
So:
P (A/B)
P(B) }
[P(B/A)]
= P(A)
This is
Bayes’ theorem
}
updating
factor
1
Linguistic Chaos
Traditional Meanings of “Probability”
New Theories
Statistician’s
(Frequency
Fraction)
Bayesian
(Probability)
Mathematical
(Probability)
Fuzzy Theories
(Fuzziness)
Random
Belief
Formal Probability Ambiguity
Variability
“Personal”
Probability
“Axiomatic”
Probability
“Aleatory”
Probability
Subjective
Probability
Vagueness
“Objective”
Probability
Uncertainty
Ill defined
Stochastic
Ontological
Confidence
“In the World”
Probability
Epistemic
Probability
Reliability
Forensic
Probability
Chance
Plausibility
Risk
Credibility
Possibility Theory Dempster Shafer
(Belief)
Unclarity
“Evidence-Based”
Probability
2
Kaplan’s
Communication Theorems
• Theorem 1: 50% of the problems in the world
result from people using the same words with
different meanings.
• Theorem 2: The other 50% comes from people
using different words with the same meaning.
Derivation of Bayes’ Theorem
1.
Establish Credibility Scale
2.
Calibrate Scale
C
0
0.5
1.0
C(A|E) = 1.0
A must be True, Given E
C(A|E) = 0 ⇔ A must be False, Given E
C(A|E) = 0.5 ⇔ E not Relevant to A
⇔
3.
Discover Properties of Scale (Resulting From Calibration)
C(A|E) + C(A|E) = 1.0
C(A+E) = C(A) C(E|A)
= C(E) C(A|E)
Therefore, C(E) C(A|E) = C(A) C(E|A)
4.
Bayes’ Theorem
Likelihood Function
 C ( E | A) 
C ( A | E ) = C ( A)

 C (E ) 
Posterior
Prior
Correction Factor
3
Understanding Bayes’ Theorem
P( A^ B ) = P( A) P( B | A) = P( B ) P( A | B )
P ( A) P ( B | A)
P( A | B) =
P( B)
P( H ) P( B | H )
Change : P( H | B ) =
P( B)
A → H
 P( E | H ) 
Change : P ( H | E ) = P ( H ) 

B → E
 P( E ) 
 C ( EH ) 
Change : C ( H | E ) = C ( H ) 

P → C
 C(E) 
Monte Hall Problem
Initially : C ( H 1) = C ( H 2) = C ( H 3) = 1
 C (E | H ) 
Bayes : C (H | E ) = C ( H ) 

 C(E) 
3
Evidence: E = Not behind door 2
1 
 C (E | H 1 ) 
C (H 1 | E ) = C ( H 1) 
= 1 / 3 2  = 1

3
1 
 C (E ) 
 2
 0 
 C (E | H 2 ) 
1
C (H 2 | E ) = C ( H 2 ) 
 = 31  = 0


 C (E ) 
 2
 1 

 = 2
3
1 
 2
C ( E ) = C ( H 1) C ( E / H 1) + C ( H 2 ) C ( E / H 2 ) + ( C ( H 3 ) C ( E / H 3 )
 C (E | H 3 ) 
C (H 3 | E ) = C ( H 3 ) 
 =
 C (E ) 
(
) (
) (
= 1 × 1 + 1 × 0 + 1 ×1
3
2
3
3
= 1 + 1 =1 2
6
3
1
3
)
4
Sampling Problem, Examples of
Bayes' Theorem with Uniform Prior
12
CURVE 0: the "prior"
CURVE 1: m=1, k=0
Probability Density (A)
10
CURVE 2: m=2, k=0
CURVE 3: m=5, k=0
8
CURVE 4: m=10, k=0
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Lambda
5
Using Information from Precursor
and Near-Miss Events
6
Bayes Theorem
For
Updating Dose/Response Curves
p1( K ) =
P0 ( K ){C K ( D)^ S } × {[1 − C K ( D)]^ ( N − S )}
∑ ( Numerator )
7
1.0
Fig.11
A Probabilistic Family of
Dose Response curve
...
Response
C1
C2
CR
k
E = {D p N p S i }
k
p1( K ) =
P0 ( K ){C K ( D)^ S } × {[1 − C K ( D)]^ ( N − S )}
∑ ( Numerator )
0
Dose
Relation of Terrorism Risk and “Ordinary Risk” Scenarios
Success Scenario, So
…
Failure Scenarios
IE
…
End States
ES 1
Terrorist Attack Scenario
ES2
IE=TES
…
Si
ES3
ESN
ES1
Attack
Preparations
ES2
8
The Fundamental Principles of Logic seen as
Special Cases of Bayes’ Theorem
 p( B | A) 
p( A | B ) = p( A)

 p( B) 
1: Modus Ponens
(Syllogism of Aristotle)
if A then B p( B | A) = 1.0
A
p ( A) = 1 = p ( A | B)
⇒ B
⇒ p( B) = 1
2: Modus Tollens
(Reductio ad absurdum)
if A then B p( B | A) = 1
NOT B
p( B) = 0
⇒ NOT A
⇒ p ( A) = 0
⇒
⇒
3: Plausible reasoning
p ( B | A) = 1
if A then B
A more likely given B ⇒ p ( A | B ) ≥ p ( A)
4: B is unlikely
Except when A is true
A is much more likely
given B
p(B) is small
p(B|A) is sizeable
p( A | B) >> p( A)
9