Models for Inexact Reasoning Models for Inexact Reasoning Reasoning - - PowerPoint PPT Presentation

Models for Inexact Reasoning Models for Inexact Reasoning Reasoning with Certainty Factors: The MYCIN Approach The MYCIN Approach

Miguel García Remesal Department of Artificial Intelligence mgremesal@fi.upm.es

The MYCIN Approach The MYCIN Approach

• Developed in 1970 by Shortliffe & Buchanan
• Developed in 1970 by Shortliffe & Buchanan

(Stanford University)

• Focused on the Medical Domain

– Selection of Therapies for Infectious Blood Diseases (Meningitis, Septicemia, etc.)

• Rule‐Based System (Backward Chaining)

y ( g)

• Use of Heuristics (“Rules of the Thumb”)
• Only theoretical success
• Only theoretical success

– Never was used in clinical practice

Overview of the Inference Process Overview of the Inference Process

• Goal: Test a hypothesis using a set of rules and
• Goal: Test a hypothesis using a set of rules and

facts (MYCIN KB) B k d h i i i f

• Backward‐chaining inference process

– Use of an inference tree

f di d li h ( G)

• Inference Tree: a directed acyclic graph (DAG)

– Nodes: facts and hypotheses – Edges: rules

• Facts are not deductible
• Hypothesis are deductible from facts and other

hypothesis using rules

Inference Tree Inference Tree

H

• Rules

R4

Rules

– R1: IF E1 AND E2 THEN H1

H2

– R2: IF H1 THEN H2 – R3: IF E3 THEN H2

R2

– R4: IF H2 THEN H

H1 R1 R3 E1 E2 E3

Rules in MYCIN Rules in MYCIN

l i i l i f

• Rules in MYCIN involve certainty factors to

deal with uncertain knowledge IF

The stain of the organism is Gram negative, AND The stain of the organism is Gram negative, AND The morphology of the organism is rod, AND The aerobicity of the organism is aerobic The aerobicity of the organism is aerobic

THEN

h l d ( ) h h There is strongly suggestive evidence (0.8) that the class of the organism is Enterobacteriaceae

Certainty Factors Certainty Factors

• Certainty factors (rules)

– Degree of confirmation (disconfirmation) of a hypothesis given concrete evidence Example (in the previous slide) – Example (in the previous slide)

• Certainty factors (evidence)

Degree of belief (disbelief) associated to a given piece – Degree of belief (disbelief) associated to a given piece

• f evidence

– Example: Example:

• CF(stain=gram‐negative) = 0.4
• CF(morphology=rod) = 0.6

CF( bi it bi ) 0 4

• CF(aerobicity=aerobic) = ‐0.4

Certainty Factors Certainty Factors

Belief 1 0.7 Total Almost total 0.5 Moderate

[ 1,1] CF ∈ −

Unknown

• 0.5

Moderate

• 0.7
• 1

Total Almost total Disbelief

Certainty Factors Certainty Factors

• Given a rule [evidence hypothesis] its CF

can be defined as follows: ( , ) ( , ) ( , ) CF h e MB h e MD h e = −

• MB(h,e): Relative measure of increased belief

( , ) ( , ) ( , ) ( , )

• MD(h,e): Relative measure of increased

di b li f disbelief

Measure of Increased Belief Measure of Increased Belief

• Relative measure of increased belief in

hypothesis h resulting from the observation of yp g evidence e

• There is an increased belief in h if P(h|e) > P(h)
• There is an increased belief in h if P(h|e) > P(h)
• Otherwise MB(h,e) = 0

Increase in the probability of h after introducing evidence e

( | ) ( ) ( , ) 1 ( ) P h e P h MB h e P h − =

R i i i i th “

1 ( ) P h −

Remaining increase in the “a priori” probability of h to reach total certainty

Measure of Increased Disbelief Measure of Increased Disbelief

• Relative measure of increased disbelief in

hypothesis h resulting from the observation of yp g evidence e

• There is an increased disbelief in h if P(h|e) <
• There is an increased disbelief in h if P(h|e) <

P(h)

Decrease in the probability of h

• Otherwise MD(h,e) = 0

Decrease in the probability of h after introducing evidence e

( ) ( | ) ( , ) P h P h e MD h e − = ( , ) ( ) P h

“A priori” probability of the hypothesis

Certainty Factors Certainty Factors

• A positive CF indicates that the evidence

supports (totally or partially) the hypothesis pp ( y p y) yp

– i.e. MB > MD

• A negative CF indicates that the evidence

discards (totally or partially) the hypothesis

– i e MD > MB i.e. MD > MB

Soundness Properties Soundness Properties

• There cannot be a

MB MD > → =

• There cannot be a

simultaneous belief and disbelief in an

MB MD MD MB > → = > → =

disbelief in an hypothesis

• The evidence e

( , ) (~ , ) CF h e CF h e + =

The evidence e supporting a given hypothesis h disfavours

( , ) 1

n i

CF h e ≤

∑

its negation to an equal extent

1 i=

• Hypotheses must be

mutually exclusive

Mutually Exclusive Hypothesis Mutually Exclusive Hypothesis

Assignment 1 Assignment 2 Assignment 3 Winner=Claws 0.8 0.8 0.7 Winner=Raven 0.7 0.2 Winner=Rusty 0.9 ‐0.4 2.4 1.0 0.3

{ }

( , ) CF winner i e =

∑

2.4 1.0 0.3

{ }

, , i claws raven rusty ∈

Inference Inference

• Firing a rule involves the use of two different

CFs:

– The CF associated to the antecedent of the rule (premises) (premises) – The CF associated to the rule

E H

( ) CF E

( ) CF R

¿ ( )? CF H ( ) CF E ¿ ( )?

R

CF H

Certainty in Compound Rules Certainty in Compound Rules

• What happens if the rule involves several

premises linked using standard connectives p g (AND, OR)?

1 2 ( ) 1 2 1 2

: ( ) min( ( ), ( ), , ( ))

n CF R

R e e e h CF e e e CF e CF e CF e ∧ ∧ ∧ ⎯⎯⎯ → ∧ ∧ ∧ = …

1 2 1 2

( ) min( ( ), ( ), , ( ))

n n

CF e e e CF e CF e CF e R h ∧ ∧ ∧ … …

1 2 ( ) 1 2 1 2

: ( ) max( ( ), ( ), , ( ))

n CF R n n

R e e e h CF e e e CF e CF e CF e ∨ ∨ ∨ ⎯⎯⎯ → ∨ ∨ ∨ = … … …

Certainty Propagation Certainty Propagation

• Calculation of the CF associated to the

consequent of a rule (after firing the rule) q ( g )

: R e h →

( )

: ( ) ( ) ( ) ( )

CF R

R e h CF e CF h CF e CF R ⎯⎯⎯ → > → = ⋅ ( ) ( ) ( ) ( ) ( ) ( ) CF e CF h CF e CF R CF e CF h > → = ⋅ ≤ → = ( ) ( )

Certainty Accumulation Certainty Accumulation

• What happens when two or more rules with the

same consequent are fired?

• How do we calculate the accumulated CF associated

to H1?

1

H1

1 1 2 1 ( )

:

CF R

R E E H ∧ ⎯⎯⎯ →

R1 R2

1 2

1 1 2 1 ( ) 2 3 4 1 ( )

: :

CF R CF R

R E E H → ∧ ⎯⎯⎯ →

E E E E E1 E2 E3 E4

Certainty Accumulation Certainty Accumulation

• Accumulation of CFs with the same sign:

1

1

( )

R

CF H x =

1 2

1 1

( )

R R

CF H y =

1 2

1

( ) ( )

R R

CF H x y x y

+

= + − ⋅

Certainty Accumulation Certainty Accumulation

• Accumulation of CFs with different signs

1

1

( )

R

CF H x =

2

1

( )

R

CF H y = x y +

1 2

1

( ) 1 min( , )

R R

x y CF H x y

+

+ = −

Example Example

• R1: IF [period holding driver license = between two and three years] THEN
• R1: IF [period_holding_driver_license = between_two_and_three_years] THEN

(0.5) [senior_driver = yes]

• R2: IF [period_holding_driver_license = more_than_three_years] THEN (0.9)

[senior_driver = yes]

• R3: IF [driving time

between 2 and 3 hours] THEN (0 5) [tired yes]

• R3: IF [driving_time = between_2_and_3_hours] THEN (0.5) [tired = yes]
• R4: IF [driving_time = more_than_4_hours] THEN (1) [tired = yes]
• R5: IF [senior_driver = yes] AND [traveling_alone = no] THEN (‐0.5)

[responsible_for_the_accident = yes]

• R6: IF [tired = yes] THEN (0.5) [responsible_for_the_accident = yes]
• R7: IF [alcohol = yes] AND [young = yes] THEN (0.7) [responsible_for_the_accident

= yes]

• Facts for driver John Doe:

– period_holding_driver_license: 2 years – driving_time: 30 minutes – traveling alone: no – traveling_alone: no – alcohol: yes CF(alcohol = yes) = 0.5 – 32 years old CF(young = yes) = 0.4

Example: Inference Tree Example: Inference Tree

responsible for the accident = p _ _ _ yes

R5 R6 R7

• 0.5

0.5 0.7

senior_driver = yes traveling_alone = no tired = yes alcohol = yes young = yes

R R

yes no

R R

y y y g y

0 5 0 9 0 5 1 0

R1 R2 R3 R4

0.5 0.9 0.5 1.0

period_holding_driver_license = between_2_and_3_years period_holding_driver_license = more_than_3_years driving_time = between_2_and_3_hours driving_time = more_than_4_hours

Example Example

• The resulting CF is very close to 0:
• The resulting CF is very close to 0:

– MYCIN cannot determine whether or not John Doe is responsible for the accident (unknown) Doe is responsible for the accident (unknown)

• Let us make inference for the other driver

involved in the accident: Jane Smith involved in the accident: Jane Smith

• Facts for driver Jane Smith:

period holding driver license: 1 year – period_holding_driver_license: 1 year – driving_time: 2 hours traveling alone: yes – traveling_alone: yes – alcohol: yes CF(alcohol = yes) = 0.5 20 years old CF(young = yes) = 0 5 – 20 years old CF(young = yes) = 0.5