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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Uncertain Logical Gates in Possibilistic Networks An Application to Human Geography

Didier Dubois1 Giovanni Fusco2 Henri Prade1 Andrea G. B. Tettamanzi2

1) IRIT, France 2) Univ. Nice Sophia Antipolis, France

SUM 2015, Qu´ ebec

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Motivations and Objectives

Possibilistic networks offer a qualitative approach for modeling epistemic uncertainty. Their practical implementation requires the specification of conditional possibility tables, as in the case of Bayesian networks for probabilities. ⇒ Develop possibilistic counterparts of noisy probabilistic connectives (and, or, max, min, . . . ).

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Probabilistic Networks with Independent Causal Influences

Definition (Noisy function) P(Y , Z1, . . . , Zn, X1, X2, . . . , Xn) = P(Y , Z1, . . . , Zn)·

n

• i=1

P(Zi | Xi), P(Y , Z1, . . . , Zn) = 1, Y = f (Z1, Z2, . . . , Zn); 0,

• therwise.

ICI Assumption P(y | x1, . . . , xn) =

• z1,...,zn:y=f (z1,...,zn)

n

• i=1

P(zi | xi).

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Canonical ICI Models

Specific choice of the function f If all variables are Boolean,

f = ∨ noisy OR f = ∧ noisy AND

If the range of the Zi’s and Y is a totally ordered set,

f = max noisy MAX f = min noisy MIN

Leaky model: introduce a leak variable Zℓ, such that P(Y , Z1, . . . , Zn, Zℓ, X1, . . . , Xn) = P(Y , Z1, . . . , Zn)·P(Zℓ)·

n

• i=1

P(Zi | Xi), P(y | x1, . . . , xn) =

• z1,...,zn,zℓ:y=f (z1,...,zn,zℓ)

P(zℓ) ·

n

• i=1

P(zi | xi).

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Possibilistic Networks with ICI

Definition (Uncertain Function) π(y | x1, . . . , xn) = max

z1,...,zn:y=f (z1,...,zn) ∗i=1,...,nπ(zi | xi),

π(y | x1, . . . , xn) =

• 1

if y = f (x1, . . . xn);

• therwise.

Leaky ICI Model π(y | x1, . . . , xn) = max

z1,...,zn,zℓ:y=f (z1,...,zn,zℓ) ∗i=1,...,nπ(zi | xi) ∗ π(zℓ).

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates

Uncertain OR Gate

π(Zi|Xi) xi ¬xi zi 1 ¬zi κi 1 Variables are Boolean (i.e., Y = y or ¬y, etc.). f (Z1, . . . , Zn) = n

i=1 Zi.

Causes may fail to produce their effects Zi = ¬zi ⇔ Xi = xi did not cause Y = y due to some inhibitor. Then we must define π(zi | xi) = 1 and π(¬zi | xi) = κi < 1. π(zi | ¬xi) = 0, since when Xi is absent, it does not cause y.

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates

Conditional Table for the Uncertain OR Gate

π(y | X1, . . . , Xn) = max

z1,...,zn:z1∨···∨zn=1 ∗n i=1π(zi | Xi)

=

n

max

i=1 π(zi | Xi) ∗ (∗j=i max(π(zj | Xj)π(¬zj | Xj));

π(¬y | X1, . . . , Xn) = max

z1,...,zn:z1∨···∨zn=0 ∗n i=1π(zi | Xi)

= π(¬z1 | X1) ∗ · · · ∗ π(¬zn | Xn). For n = 2 π(y | X1X2) x1 ¬x1 x2 1 1 ¬x2 1 π(¬y | X1X2) x1 ¬x1 x2 κ1 ∗ κ2 κ2 ¬x2 κ1 1

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates

Uncertain AND Gate

π(Zi|Xi) xi ¬xi zi 1 ¬zi κi 1 Same local conditional table as OR However, Xi = xi is now a necessary cause for Y = y. f (Z1, . . . , Zn) = n

i=1 Zi.

For n = 2 π(y | X1X2) x1 ¬x1 x2 1 ¬x2 π(¬y | X1X2) x1 ¬x1 x2 max(κ1, κ2) 1 ¬x2 1 1

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates

Uncertain OR and AND Gates with Leak

Leaky OR, for n = 2 π(y | X1X2) x1 ¬x1 x2 1 1 ¬x2 1 κℓ π(¬y | X1X2) x1 ¬x1 x2 κ1 ∗ κ2 κ2 ¬x2 κ1 1 Leaky AND, for n = 2 π(y | X1X2) x1 ¬x1 x2 1 κL ¬x2 κL κL π(¬y | X1X2) x1 ¬x1 x2 max(κ1, κ2) 1 ¬x2 1 1 The 0 entries have been replaced by the leakage coefficient.

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates

Generation of a TPC through an Uncertain OR logical gate

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Comparison with Probabilistic Gates

Uncertain OR Gates vs. Noisy OR Gates

If ∗ = min, two important differences of behavior of the Uncertain OR appear:

1 the presence of two or more causes does not reinforce the

certainty of the effect wrt the presence of the most influential cause;

2 two or more causes that are individually insufficient to make

an effect plausible are still insufficient to make it plausible even if joined together. Uncertain gates are less expressive than noisy gates.

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain MAX and MIN Gates

Uncertain MAX

Multiple-valued extension of the uncertain OR Y (hence the Zi) are valued on severity scale L = {0 < 1 < · · · < m} Y = max(Z1, . . . , Zn) π(y | x1, . . . , xn) = max

z1,...,zn:y=max(z1,...,zn) ∗n i=1π(zi | xi)

=

n

max

i=1 π(Zi = y | xi) ∗ (∗j=iΠ(Zj ≤ y | xj)) .

Possibility Table for for 3 levels of strength 0, 1, 2 π(Zi | Xi) Xi = 2 Xi = 1 Xi = 0 Zi = 2 1 Zi = 1 κ12

i

1 Zi = 0 κ02

i

κ01

i

1

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain MAX and MIN Gates

Uncertain MAX: Global Conditional Possibility Table

π(Y = j|x) =

n

max

i=1 π(Zi = j|xi) ∗ (∗ℓ=iΠ(Zℓ ≤ j|xℓ))

For n = 2, m = 2, 3 levels of strength

x π(2 | x) π(1 | x) π(0 | x) (2, 2) 1 max(κ12

1 , κ12 2 )

κ02

1 ∗ κ02 2

(2, 1) 1 1 κ02

1 ∗ κ01 2

(2, 0) 1 κ12

1

κ02

1

(1, 2) 1 1 κ01

1 ∗ κ02 2

(1, 1) 1 κ01

1 ∗ κ01 2

(1, 0) 1 κ01

1

(0, 2) 1 κ12

2

κ02

2

(0, 1) 1 κ01

2

(0, 0) 1

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain MAX and MIN Gates

Uncertain MIN

π(y | x1, . . . , xn) = max

z1,...,zn:y=min(z1,...,zn) ∗n i=1π(zi | xi)

=

n

max

i=1 π(Zi = y | xi) ∗ (∗j=iΠ(Zj ≥ y|xj)).

For n = 2, m = 2, 3 levels of strength

x π(2|x) π(1|x) π(0|x) (2, 2) 1 max(κ12

1 , κ12 2 )

max(κ02

1 , κ02 2 )

(2, 1) 1 max(κ02

1 , κ01 2 )

(2, 0) κ12

1

1 (1, 2) 1 max(κ01

1 , κ02 2 )

(1, 1) 1 max(κ01

1 , κ01 2 )

(1, 0) 1 (0, 2) κ12

2

1 (0, 1) 1 (0, 0) 1

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Algorithm 1. uncertain-MAX(Y , prm)

1: π(Y |X1, . . . , Xn) ← 0 2: for all x ∈ X1 × . . . × Xn do 3:

K ← {k : condi, k ∈ prm, x | = condi} {Select the parameters that apply to x}

4:

for all y = (y1, . . . , yK) ∈ Y K do

5:

β ← mini=1,...,K{κiyi}

6:

¯ y ← maxi=1,...,K{yi}

7:

π(¯ y | x) ← max{β, π(¯ y | x)}

8:

end for

9: end for 10: return π(Y | X1, . . . , Xn)

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Uncertain MAX with Threshold

The simultaneous presence of individually “weak” causes:

Noisy MAX: may lead to a plausible effect Uncertain MAX: may not lead to a plausible effect

Yet such situations arise in applications and are fully compatible with possibility theory! We propose the Uncertain MAX with Threshold

usual parameters of an uncertain MAX: threshold θj ∈ {1, 2, . . .} for each value yj of Y if at least θj causes concur, Π(Y = yj | x) = 1

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Algorithm 2. uncertain-MAX-threshold(Y , prm, thr)

1: π(Y |X1, . . . , Xn) ← 0 2: κℓ ← 0 3: for all condi, k ∈ prm : condi = ⊤ do 4:

κℓ ← max{κℓ, k}

5: end for 6: for all x ∈ X1 × . . . × Xn do 7:

cnt ← 0 {A vector of counters, one for each y ∈ Y }

8:

K ← {k : condi, k ∈ prm, x | = condi} {Select the params that apply to x}

9:

for all y = (y1, . . . , yK) ∈ Y K do

10:

β ← mini=1,...,K{κiyi }

11:

¯ y ← maxi=1,...,K{yi}

12:

if β > κℓ(¯ y) then

13:

cnt¯

y ← cnt¯ y + 1

14:

end if

15:

if cnt¯

y ≥ θ¯ y then

16:

β ← 1

17:

end if

18:

π(¯ y | x) ← max{β, π(¯ y | x)}

19:

end for

20: end for 21: return π(Y | X1, . . . , Xn)

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

An Application to Human Geography

This work is has been motivated by a collaboration with the geographers of the ESPACE laboratory in Nice Modeling social polarization in the Aix-Marseille area Valorization/devalorization of the residential space at the town scale Existing 26-variable Bayesian network based on expert knowledge We constructed the corresponding possibilistic network T −1

1

probability-to-possibility converse transformation

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

The BN model for the valorization/devalorization of municipalities in the study area

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Results

Both the BN and the PN were used to produce trend scenarios for social polarization in the 439 municipalities of the Aix-Marseille metropolitan area. Target variable: Situation T2 ∈ {V , D, O}. The BN predicts a single most probable value for all municipalities (fallacious impression of certainty) The min-max PN produces in many cases sets of fully possible (Π = 1) values (= complete ignorance) BN and PN tend to agree (∼ 75%), but the PN incorporates more uncertainty PN assigns complete ignorance to 130 municipalities out of 439 (∼ 30%) vs. 85 for BN w/ 0.25 “tolerance”

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Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion

Conclusions and Future Work

First detailed study of the possibilistic counterpart of noisy gates Application to human geography Interesting for the practical use of possibilistic networks Noticeable differences of behavior have been revealed Generally speaking, PNs appear to be more cautious. Future work:

Compare expressive power of BNs and PNs Develop a complete panoply of uncertain gates

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