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) • Backward ‐ chaining inference process B k d h i i i f – Use of an inference tree • Inference Tree: a directed acyclic graph (DAG) f di d li h ( G) – 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 R 4 • Rules Rules H 2 – R 1 : IF E 1 AND E 2 THEN H 1 – R 2 : IF H 1 THEN H 2 R 2 – R 3 : IF E 3 THEN H 2 H 1 – R 4 : IF H 2 THEN H R 1 R 3 E 1 E 2 E 3

Rules in MYCIN Rules in MYCIN • Rules in MYCIN involve c ertainty factors to l i i l i f 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 There is strongly suggestive evidence (0.8) that the h l d ( ) h h 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 of evidence – Example: Example: • CF(stain=gram ‐ negative) = 0.4 • CF(morphology=rod) = 0.6 • CF(aerobicity=aerobic) = ‐ 0.4 CF( bi it bi ) 0 4

Certainty Factors Certainty Factors Belief 1 Total 0.7 Almost total 0.5 Moderate CF ∈ − [ 1,1] 0 Unknown Moderate -0.5 Almost total -0.7 -1 Total Disbelief

Certainty Factors Certainty Factors • Given a rule [ e vidence � h ypothesis ] 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 − ( | ) ( ) P h e P h = ( , ) MB h e − 1 1 ( ) ( ) P h P h R Remaining increase in the “a i i i i th “ 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 Decrease in the probability of h • Otherwise MD ( h , e ) = 0 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 • There cannot be a simultaneous belief and > > → → = = 0 0 0 0 disbelief in an disbelief in an MB MB MD MD hypothesis > → = 0 0 MD MB • The evidence e The evidence e + = supporting a given ( , ) (~ , ) 0 CF h e CF h e hypothesis h disfavours n its negation to an equal ∑ CF h e ≤ ( , ) 1 extent i i = 1 • Hypotheses must be mutually exclusive

Mutually Exclusive Hypothesis Mutually Exclusive Hypothesis Assignment Assignment Assignment 1 2 3 Winner=Claws 0.8 0.8 0.7 Winner=Raven 0.7 0.2 0 Winner=Rusty 0.9 0 ‐ 0.4 ∑ = ( , ) CF winner i e 2.4 2.4 1.0 1.0 0.3 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 R ¿ ¿ ( ( )? )? ( ) ( ) CF CF H H CF E CF E R

Certainty in Compound Rules Certainty in Compound Rules • What happens if the rule involves several premises linked using standard connectives p g (AND, OR)? ∧ ∧ ∧ ⎯⎯⎯ → … : R e e e h 1 2 n ( ) CF R ∧ ∧ ∧ ∧ ∧ ∧ = … … ( ( ) ) min( min( ( ), ( ), ( ( ), ), , , ( ( )) )) CF e CF e e e e e CF e CF e CF e CF e CF e CF e 1 1 2 2 1 1 2 2 n n ∨ ∨ ∨ ⎯⎯⎯ → … : R e R e e h h 1 2 n ( ) CF R ∨ ∨ ∨ = … … ( ) max( ( ), ( ), , ( )) CF e e e CF e CF e CF e 1 2 1 2 n n

Certainty Propagation Certainty Propagation • Calculation of the CF associated to the consequent of a rule (after firing the rule) q ( g ) ⎯⎯⎯ → → : : R e R e h h ( ) CF R > > → → = = ⋅ ⋅ ( ) ( ) 0 0 ( ) ( ) ( ) ( ) ( ) ( ) CF e CF e CF h CF h CF e CF e CF R CF R ≤ → = ( ) ( ) 0 ( ) ( ) 0 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 H 1 ? 1 H 1 ∧ ⎯⎯⎯ → → : : R E E H 1 1 1 1 2 2 1 1 ( ( ) ) CF R CF R 1 ∧ ⎯⎯⎯ → : R 1 R 2 R E E H 2 3 4 1 ( ) CF R 2 E E 1 E E 2 E E 3 E E 4

Certainty Accumulation Certainty Accumulation • Accumulation of CFs with the same sign: = ( ) CF H x 1 1 R R 1 1 = ( ) CF H y 1 R 2 = + − ⋅ ( ) ( ) CF H x y x y + 1 R R 1 2

Certainty Accumulation Certainty Accumulation • Accumulation of CFs with different signs = ( ) CF H x 1 R 1 = ( ) CF H y 1 R 2 + + x x y y = − ( ) CF H + 1 R R 1 min( , ) 1 2 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 R3: IF [driving_time = between_2_and_3_hours] THEN (0.5) [tired = yes] 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 R 5 R 6 R 7 -0.5 0.5 0.7 senior_driver = traveling_alone = tired = yes y alcohol = yes y young = yes y g y yes yes no no R R 1 R R 2 R R 3 R 4 R 0 5 0.5 0 9 0.9 0 5 0.5 1 0 1.0 period_holding_driver_license = period_holding_driver_license = driving_time = driving_time = between_2_and_3_years more_than_3_years between_2_and_3_hours 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