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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

l d b d d d i h

• Developed by Duda and Hart during the 70s at

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

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• Rule‐based system (backwards chaining)
• 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 (LSe) – Necessity (LNe) – Credibility (O)

• All these factors are provided by experts

p y p

e h (LS, LN) e 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 ( ) ( ) ( ) ( ) 1 ( ) P h P h O h P h P h = = − ( ) ( )

Evolution of the Credibility Factor Evolution of the Credibility Factor

l i f “ i i” d id

• Evolution of “a priori” CFs due to evidence

– In the case of e to be true

( | ) ( )

e

O h e LS O h = ⋅

– In the case of e to be false In the case of having n independent evidences

( | ) ( )

e

O h e LN O h = ⋅

– In the case of having n independent evidences

• ver the hypothesis h

1 2 3

1 2 3

( | , , , , ) ( )

n

n e e e e

O h e e e e LS LS LN LS O h = ⋅ ⋅ ⋅ ⋅ ⋅ K K

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 ( | ) ( | )

e

P e h LS 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 ( | ) ( | )

e

P e h LN P e h = ( | )

Interpretation of the Degree of Sufficiency Interpretation of the Degree of Sufficiency

LS Effects over the implication [e h]

h is false when e holds

• r

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 ∞

• r

if e holds then h holds

Interpretation of the Degree of Necessity Interpretation of the Degree of Necessity

LN Effects over the implication [e h]

h is false when e does not hold

• r

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 ē i f bl f h t h ld 1 << LN ē is favorable for h to hold ∞ ē is sufficient for h to hold

Inference Process Inference Process

• Given a concrete case E including:

– Observed probabilities P(ei|E) for facts e1,…,ei 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 e E P h e P e E = ⋅ + ⋅ ( | ) ( | ) ( | ) ( | ) ( | ) P h E P h e P e E P h 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

h l i h i f “li ” i f i

• The latter is the equation of a “line” satisfying

the following conditions:

[ ( | ) 0] [ ( | ) ( | )] [ ( | ) ( )] [ ( | ) ( )] P e E P h E P h e P e E P e P h E P h = ↔ = = ↔ =

h “l ” l (b l

[ ( | ) ( )] [ ( | ) ( )] [ ( | ) 1] [ ( | ) ( | )] P e E P e P h E P h P e E P h E P h e ↔ = ↔ =

• This “line” is not a line (but a piecewise linear

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 PR(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:

( | ) h ( | ) ( | ) 1 ( | ) O h e P h e O h e = + ( | ) ( | ) ( | ) O h e P h e = ( | ) 1 ( | ) P h e 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 [ ( ) (h)] [1 (h| )]

• t [p(e), p(h)], [1, p(h|e)]
• After that, we calculate PR(h|E) using the

corresponding equation and the value P(e|E)

Inference Process Inference Process

• From PR(h|E) we calculate OR(h|E)

( | ) ( | ) 1 ( | )

R R

P h E O h E P h E = −

• And from O (h|E) we calculate L

1 ( | )

R

P h E And from OR(h|E) we calculate LR

( | ) O h E ( | ) ( )

R R

O h E L 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:

1 2

( | ) ( )

n

R R R

O h E L L L O h = ⋅ ⋅ ⋅ ⋅ K

• Then, we calculate P(h|E) as follows:

( | ) ( | ) 1 ( | ) O h E P h E O h E = + 1 ( | ) 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)?