Models for Inexact Reasoning Reasoning with Subjective Pseudo - PowerPoint PPT Presentation

Models for Inexact Reasoning Reasoning with Subjective Pseudo Reasoning with Subjective Pseudo ‐ Probabilities: The PROSPECTOR Approach Miguel García Remesal Department of Artificial Intelligence mgremesal@fi.upm.es

The PROSPECTOR System The PROSPECTOR System • Developed by Duda and Hart during the 70s at l d b d d d i h SRI International • Intended user is an exploration geologist in the early stages of investigating a possibly y g g g p y drilling site • Successfully used in practice: Successfully used in practice: – Helped to discover an important deposit of Molybdenum worth more than $100M in the Molybdenum worth more than $100M in the State of Washington (USA)

The PROSPECTOR Approach The PROSPECTOR Approach • Rule ‐ based system (backwards chaining) l b d (b k d h i i ) • Use of an inference tree to support the pp reasoning process (similarly to MYCIN) • PROSPECTOR uses probabilities to represent PROSPECTOR uses probabilities to represent uncertain knowledge – A priori “A priori” • Independent of the case • Independent of the problem to be solved (the • Independent of the problem to be solved (the hypothesis) – “A posteriori” A posteriori • Evidence for a concrete case (data dependent)

Sufficiency, Necessity and Credibility Factors • PROSPECTOR uses three measures to represent the conditional probabilities p p involved in an implication: – Sufficiency (LS ) – Sufficiency (LS e ) – Necessity (LN e ) – Credibility (O) • All these factors are provided by experts p y p (LS, LN) e e h h OH

Credibility Factor Credibility Factor • Given a hypothesis h, PROSPECTOR is targeted to analyze the evolution of the ratio between: y – The probability of h to hold – The probability of h to be false The probability of h to be false • This ratio is called the Credibility Factor ( ) ( ) ( ) ( ) P h P h P h P h = = − ( ) O h ( ) ( ) 1 ( ) ( ) P h P h

Evolution of the Credibility Factor Evolution of the Credibility Factor • Evolution of “a priori” CFs due to evidence l i f “ i i” d id – In the case of e to be true = ⋅ ( | ) ( ) O h e LS O h e – In the case of e to be false = ⋅ ( | ) ( ) O h e LN O h e – In the case of having n independent evidences In the case of having n independent evidences over the hypothesis h = ⋅ ⋅ ⋅ ⋅ ⋅ K K ( | , , , , ) ( ) O h e e e e LS LS LN LS O h 1 2 3 n e e e e 1 2 3 n

Sufficiency Factor • Measures the extent to which the antecedent is sufficient for the consequent to hold q – Degree to which if the premises are true then the hypothesis is also true hypothesis is also true • The higher the better (ideally ∞ ) ( | ) ( | ) P e h P e h = LS e ( | ) P e h

Necessity Factor Necessity Factor • Measures the extent to which the antecedent is necessary for the consequent to hold y q – Degree to which if the consequent is true then the premises are also true premises are also true • The smaller the better (ideally 0) ( | ) ( | ) P e h P e h = LN e ( | ( | ) ) P e h

Interpretation of the Degree of Sufficiency Interpretation of the Degree of Sufficiency Effects over the implication [ e � h ] LS h is false when e holds 0 or ē is necessary for h to hold 0 < LS << 1 e is not favorable for h to hold 1 e has no effect over h 1 << LS e is favorable for h to hold e is sufficient for h to hold ∞ or if e holds then h holds

Interpretation of the Degree of Necessity Interpretation of the Degree of Necessity Effects over the implication [ e � h ] LN h is false when e does not hold 0 or e is necessary for h to hold 0 < LN << 1 ē is not favorable for h to hold 1 ē has no effect over h 1 << LN 1 << LN ē i f ē is favorable for h to hold bl f h t h ld ∞ ē is sufficient for h to hold

Inference Process Inference Process • Given a concrete case E including: – Observed probabilities P(e i |E) for facts e 1 ,…,e i p ( i | ) 1 i – A target hypothesis h • Aim: Calculate P(h|E) • Aim: Calculate P(h|E) • Applying the total probability theorem: = ⋅ + + ⋅ ( | ( | ) ) ( | ) ( | ) ( | ( | ) ) ( | ) ( | ) ( | ( | ) ) P h E P h E P h e P h e P e E P e E P h e P h e P e E P e E b = ⋅ + ⋅ − ( | ) ( | ) ( | ) ( | ) (1 ( | )) P h E P h e P e E P h e P e E

Inference Process Inference Process • The latter is the equation of a “line” satisfying h l i h i f “li ” i f i the following conditions: = ↔ = [ ( | ) 0] [ ( | ) ( | )] P e E P h E P h e = ↔ ↔ = [ ( | [ ( | ) ) ( )] ( )] [ ( | [ ( | ) ) ( )] ( )] P e E P e E P e P e P h E P h E P h P h = ↔ = [ ( | ) 1] [ ( | ) ( | )] P e E P h E P h e • This “line” is not a line (but a piecewise linear h “l ” l (b l interpolation) why?? – The pair [P(e), P(h)] was provided by experts – It does not satisfy any mathematical conditions y y

Inference Process Inference Process • We can calculate P R (h|E) (due a single rule R) given P(e|E) using piecewise linear g ( | ) g p interpolation • First we calculate P(h|e) and P(h| ē ) as • First, we calculate P(h|e) and P(h| ē ) as follows: ( | ) ( | ) O h e h = + ( | ) P h e 1 ( | ) ( | ) O h e ( | ) O h e P h e = = + ( | ) ( | ) P h e 1 ( | ) O h e

Inference Process Inference Process • Then, we obtain the equation for line s or t depending on the following conditions: p g g – If 0 < P(e|E) < P(e) • s � [0 P(h| ē )] [p(e) p(h)] s � [0, P(h| ē )], [p(e), p(h)] – If P(e) < P(e|E) < 1 • t � [p(e), p(h)], [1, p(h|e)] t � [ ( ) (h)] [1 (h| )] • After that, we calculate P R (h|E) using the corresponding equation and the value P(e|E)

Inference Process Inference Process • From P R (h|E) we calculate O R (h|E) ( | ) P h E = − R ( | ) O h E R 1 1 ( | ( | ) ) P h E P h E R • And from O (h|E) we calculate L And from O R (h|E) we calculate L R ( | ( | ) ) O O h E h E = R L R ( ) ( ) O h O h

Inference Process Inference Process • If several rules involve the same consequent, we accumulate the beliefs contributed by the y different rules: = ⋅ ⋅ ⋅ ⋅ K ( | ) ( ) O h E L L L O h R R R 1 2 n • Then, we calculate P(h|E) as follows: ( | ) O h E = + ( | ) P h E + 1 1 ( | ( | ) ) O h E O h E

Inference Process: Example Inference Process: Example • Rules: – R1: IF (position=normal) AND (umbilical_cord=normal) THEN (20, 0.1) (birth=normal) – R2: IF (position=podalic) AND (umbilical cord=normal) THEN (5, 0.5) R2: IF (position podalic) AND (umbilical_cord normal) THEN (5, 0.5) (birth=normal) • “A priori” evidence – P(position=normal) = 0.5 – P(position=podalic) = 0.15 – P(umbilical cord=normal) = 0.85 ( _ ) – P(birth=normal) = 0.60 • A concrete case (E) – P(position=normal|E) = 0.70 – P(position=podalic|E) = 0.20 – P(umbilical cord=normal|E) = 0.20 P(umbilical_cord normal|E) 0.20 – P(birth=normal|E)?