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CHAPTER 5

Uncertainty and Complexity

 

Uncertainty

Two types of uncertainty1 are considered:

  1. Type 1 Uncertainty. This is uncertainty about the truth of the input evidences.
  2. Type 2 Uncertainty. This is uncertainty about the validity of the rules.

To deal with the first type, we can analyze the impact of varying the input probabilities, over possible or plausible ranges, on the probabilities of the conclusions. To deal with the second type, we can investigate the impact of varying both the sensitivity and the specificity of the rules on the conclusions.

Each rule has one conclusion and a combination of input evidences. The validity of each rule can be specified by two parameters:

  1. The sensitivity of the rule. This is the conditional probability P(C|E) that the conclusion is true when the rule's combined evidences are true. In the simulations that follow, P(C|E) is varied from 0% to 100% in steps of 5%.
  2. The specificity of the rule. This is the conditional probability P(-C|-E) that the conclusion is false when the rule's combined evidences are false. P(C|-E), the conditional probability that the conclusion is true when the rule's combined evidences are false, is equal to 1-P(-C|-E). In the following simulations, P(C|-E), is varied from 0% to 50% in steps of 10%.

For a rule to be absolutely valid, P(C|E) and P(C|-E) must be equal to 100% and 0%, respectively.

The sensitivity of the probability of the conclusions (C3, C4, and C5) to a 10% change in the uncertainty (sensitivity and specificity) of the rules is illustrated in Plates 5-1 and 5-2. The systematic analysis of the combined impact of three parameters (1) the strength of the evidences, (2) the sensitivity of the rules, and (3) the specificity of the rules is provided in Plates 5-3 through 5-10.

Complexity

All the input evidences are not necessarily independent. For example, a sudden reduction by the bank of the firm's line of credit can signal to other creditors a decrease in the company's creditworthiness. The use of predatory collection techniques may be encouraged by laws that favor lenders over borrowers. Such dependencies can add considerably to the complexity of the expert system. But the complexity can be handled by using Monte Carlo simulations.

 


1 See James N. Siddall, Expert Systems for Engineers, 1990, at 143-145 (Type 1 and Type 2 Uncertainty), 155-164 (Use of Subjective Probability and Probability Laws in Expert Systems), 168-175 (Bayes' Theorem in Expert Systems), and 181-183 (Discussion).   

 

SUBJECTIVE PROBABILITIES
THAT THE INPUT EVIDENCES ARE TRUE

 

 

UNCERTAINTY

TYPE 1

TYPE 2

EVIDENCES Probability P(E)
Maximum subjective probability
Strength of evidence
 
RULES   Sensitivity P(C|E)
Conditional probability that the conclusion is true when the rule's combined evidences are true
  Specificity P(-C|-E)
Conditional probability that the conclusion is false when the rule's combined evidences are false
  
Table 5-1    Types of Uncertainty

The Table lists the types of uncertainty and the kinds of subjective probabilities one must deal with when using an expert system.1 By analogy with clinical decision making, we can say:2

  1. A rule is sensitive when it has a high P(C|E) and low P(-C|E).
  2. A rule is specific when it has a low P(C|-E) and high P(-C|-E).

In clinical decision making, sensitivity and specificity refer to the true-positive rate of a test and the true-negative rate of a test, respectively.

Sources:
1 See James N. Siddall, Expert Systems for Engineers, 1990, at 143-145 (Type 1 and Type 2 Uncertainty), 155-164 (Use of Subjective Probability and Probability Laws in Expert Systems).
2 For definitions and a discussion of test sensitivity and test specificity, see, for example, Milton C. Weinstein, Harvey V. Fineberg, et al., Clinical Decision Analysis, 1980, at 84-85.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

   

 

CATEGORIES OF EFFECTS ANALYZED

 

EFFECT ANALYZED PLATE

STRENGTH OF
EVIDENCES

SENSITIVITY
OF RULES
P(C|E)

SPECIFICITY
OF RULES
P(-C|-E)

Effect of the strength of the evidences. The rules are assumed to be absolutely valid

3-1

0%-100%

100%

100%

Combined effect of the strength of the evidences and of a 10% decrease in the validity (sensitivity or specificity) of the rules

5-1
5-2

0%-100%
0%-100%

90%-100%
100%

100%
90%-100%

Combined effect of the sensitivity and specificity of the rules. The strength of the evidences is assumed to be maximum.

5-3

100%

0%-100%

50%-100%

Combined effect of the strength of the evidences and of the sensitivity of the rules. The rules are assumed to be perfectly specific.

5-4

0%-100%

50%-100%

100%

Combined effect of the strength of the evidences, the sensitivity of the rules, and the specificity of the rules.

5-5
5-6
5-7
5-8
5-9
5-10

0%-100%
0%-100%
0%-100%
0%-100%
0%-100%
0%-100%

100%
90%
80%
70%
60%
50%

50%-100%
50%-100%
50%-100%
50%-100%
50%-100%
50%-100%

  
Table 5-2    Categories of Effects Analyzed

The Monte Carlo simulations are designed to elicit the possible effects of three kinds of subjective inputs on the probability of conclusions C3, C4, and C5:

  1. The strength of the evidences.
  2. The sensitivity of the rules.
  3. The specificity of the rules.

Five categories of effects covering a wide range of uncertainty are investigated. To assess risks using the expert system:

  1. Select the strength of the evidences (Type 1 Uncertainty) that is most consistent with your own experience or judgment.
  2. Select a plausible range for the validity of the rules (Type 2 Uncertainty).
  3. Locate the appropriate Plate.
  4. Use the Charts in the Plate to extract estimates for the subjective probabilities of conclusions C3, C4, and C5. The Charts codify how the subjective input probabilities propagate through the expert system.
  5. Check if these estimates are consistent with your own experience or expectation.

Step 1 requires empirical data based on observations of the marketplace; step 2 requires expertise and insight into the inner workings of the financial services sector and related government activities. If you cannot codify the input probabilities, use the next best thing, your intuition.1

Source:
1 On the use of subjective probabilities, see James N. Siddall, Expert Systems for Engineers, 1990, at 155-164.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

   

 

SENSITIVITY ANALYSIS
EFFECT OF A 10% REDUCTION IN THE SENSITIVITY OF THE RULES ON CONCLUSIONS

 

Plate 5-1
  
Plate 5-1    Sensitivity Analysis: Effect of a 10% Reduction in the Sensitivity of the Rules on Conclusions

The sensitivity of a rule is the conditional probability P(C|E) that the conclusion is true when the rule's combined input evidences are true. Three conclusions are considered:

C3: the bank freezes, reduces, or calls the firm's loan
C4: the firm is destabilized
C5: the firm is destroyed or cannibalized.

A decrease in the sensitivity of the rules decreases the subjective probability of the conclusions (arrow a).

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SPECIFICITY ANALYSIS
EFFECT OF A 10% REDUCTION IN THE SPECIFICITY OF THE RULES ON CONCLUSIONS

 

Plate 5-2
  
Plate 5-2    Specificity Analysis: Effect of a 10% Reduction in the Specificity of the Rules on Conclusions

The specificity of a rule is the conditional probability P(-C|-E) that the conclusion is false when the rule's combined input evidences are false. Three conclusions are considered:

C3: the bank freezes, reduces, or calls the firm's loan
C4: the firm is destabilized
C5: the firm is destroyed or cannibalized.

A decrease in the specificity of the rules increases the subjective probability of the rules (arrow a).

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE VALIDITY OF THE RULES (TYPE 2 UNCERTAINTY)

 

Plate 5-3
  
Plate 5-3    Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Validity of the Rules (Type 2 Uncertainty) -- 0%--P(C|E)--100% and 0%--P(C|-E)--50%. The Strength of the Input Evidences is Assumed to be 100%.

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the validity of the rules. The validity of the rules is parametrized as follows: P(C|E) is varied from 0% to 100% in 5% steps; and P(C|-E) is varied from 0% to 50% in 10% steps (or, equivalently P(-C|-E) is varied from 100% to 50% in 10% steps). The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The subjective probabilities of environment-related (Group 2) evidences are assumed to be maximum: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE STRENGTH OF THE EVIDENCES AND TO THE VALIDITY OF THE RULES

 

Plate 5-4
  
Plate 5-4    Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Strength of the Evidences (Type 1 Uncertainty) and to the Validity of the Rules (Type 2 Uncertainty)--50%--P(C|E)--100% and P(C|-E)=0%

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the strength of the input evidences (E1, E2, and E3; E7 and E8; and E11 and E12). The validity of the rules is parametrized as follows: P(C|-E) is fixed at 0% (or, equivalently, P(-C|-E) is fixed at 100%); and P(C|E) is varied from 50% to 100% in 10% steps. The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The maximum subjective probabilities of environment-related (Group 2) evidences are assumed to be as follows: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE STRENGTH OF THE EVIDENCES AND TO THE VALIDITY OF THE RULES

 

Plate 5-5
  
Plate 5-5    Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Strength of the Evidences (Type 1 Uncertainty) and to the Validity of the Rules (Type 2 Uncertainty)--P(C|E)=100% and 0%--P(C|-E)--50%

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the strength of the input evidences (E1, E2, and E3; E7 and E8; and E11 and E12). The validity of the rules is parametrized as follows: P(C|E) is fixed at 100%; and P(C|-E) is varied from 0% to 50% in 10% steps. The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The maximum subjective probabilities of environment-related (Group 2) evidences are assumed to be as follows: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE STRENGTH OF THE EVIDENCES AND TO THE VALIDITY OF THE RULES

 

Plate 5-6
  
Plate 5-6    Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Strength of the Evidences (Type 1 Uncertainty) and to the Validity of the Rules (Type 2 Uncertainty)--P(C|E)=90% and 0%--P(C|-E)--50%

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the strength of the input evidences (E1, E2, and E3; E7 and E8; and E11 and E12). The validity of the rules is parametrized as follows: P(C|E) is fixed at 90%; and P(C|-E) is varied from 0% to 50% in 10% steps. The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The maximum subjective probabilities of environment-related (Group 2) evidences are assumed to be as follows: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE STRENGTH OF THE EVIDENCES AND TO THE VALIDITY OF THE RULES

 

Plate 5-7
  
Plate 5-7    Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Strength of the Evidences (Type 1 Uncertainty) and to the Validity of the Rules (Type 2 Uncertainty)--P(C|E)=80% and 0%--P(C|-E)--50%

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the strength of the input evidences (E1, E2, and E3; E7 and E8; and E11 and E12). The validity of the rules is parametrized as follows: P(C|E) is fixed at 80%; and P(C|-E) is varied from 0% to 50% in 10% steps. The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The maximum subjective probabilities of environment-related (Group 2) evidences are assumed to be as follows: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE STRENGTH OF THE EVIDENCES AND TO THE VALIDITY OF THE RULES

 

Plate 5-8
  
Plate 5-8    Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Strength of the Evidences (Type 1 Uncertainty) and to the Validity of the Rules (Type 2 Uncertainty)--P(C|E)=70% and 0%--P(C|-E)--50%

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the strength of the input evidences (E1, E2, and E3; E7 and E8; and E11 and E12). The validity of the rules is parametrized as follows: P(C|E) is fixed at 70%; and P(C|-E) is varied from 0% to 50% in 10% steps. The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The maximum subjective probabilities of environment-related (Group 2) evidences are assumed to be as follows: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE STRENGTH OF THE EVIDENCES AND TO THE VALIDITY OF THE RULES

 

Plate 5-9
  
Plate 5-9    Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Strength of the Evidences (Type 1 Uncertainty) and to the Validity of the Rules (Type 2 Uncertainty)--P(C|E)=60% and 0%--P(C|-E)--50%

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the strength of the input evidences (E1, E2, and E3; E7 and E8; and E11 and E12). The validity of the rules is parametrized as follows: P(C|E) is fixed at 60%; and P(C|-E) is varied from 0% to 50% in 10% steps. The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The maximum subjective probabilities of environment-related (Group 2) evidences are assumed to be as follows: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 

SENSITIVITY OF THE SUBJECTIVE PROBABILITY
TO THE STRENGTH OF THE EVIDENCES AND TO THE VALIDITY OF THE RULES

 

Plate 5-10
  
Plate 5-10   Sensitivity of the Subjective Probability that the Firm is Destabilized and Destroyed, after the Bank Tightens its Credit or Calls its Loan, to the Strength of the Evidences (Type 1 Uncertainty) and to the Validity of the Rules (Type 2 Uncertainty)--P(C|E)=50% and 0%--P(C|-E)--50%

The subjective probabilities P(C3), P(C4), and P(C5) are computed as a function of the strength of the input evidences (E1, E2, and E3; E7 and E8; and E11 and E12). The validity of the rules is parametrized as follows: P(C|E) is fixed at 50%; and P(C|-E) is varied from 0% to 50% in 10% steps. The subjective probabilities of firm-related (Group 1) evidences are assumed to be fixed. The maximum subjective probabilities of environment-related (Group 2) evidences are assumed to be as follows: P(E1)=100%, P(E2)=90%, and P(E3)=80%; P(E7)=90% and P(E8)=80%; P(E11)=90% and P(E12)=80%. Coordinates in decimal form (1=100%); tabulated data in percent.

[Copyright -- 1998 by MACROKNOW INC. All rights reserved.]

  

 











 

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