Uncertain Logical Gates in Possibilistic Networks An Application to - PowerPoint PPT Presentation

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain Logical Gates in Possibilistic Networks An Application to Human Geography Didier Dubois 1 Giovanni Fusco 2 Henri Prade 1 Andrea G. B. Tettamanzi 2 1 ) IRIT, France 2 ) Univ. Nice Sophia Antipolis, France SUM 2015, Qu´ ebec 1 / 21

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, . . . ). 2 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Probabilistic Networks with Independent Causal Influences Definition (Noisy function) n � P ( Y , Z 1 , . . . , Z n , X 1 , X 2 , . . . , X n ) = P ( Y , Z 1 , . . . , Z n ) · P ( Z i | X i ) , i =1 � 1 , Y = f ( Z 1 , Z 2 , . . . , Z n ); P ( Y , Z 1 , . . . , Z n ) = 0 , otherwise. ICI Assumption n � � P ( y | x 1 , . . . , x n ) = P ( z i | x i ) . i =1 z 1 ,..., z n : y = f ( z 1 ,..., z n ) 3 / 21

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 Z i ’s and Y is a totally ordered set, f = max noisy MAX f = min noisy MIN Leaky model: introduce a leak variable Z ℓ , such that n � P ( Y , Z 1 , . . . , Z n , Z ℓ , X 1 , . . . , X n ) = P ( Y , Z 1 , . . . , Z n ) · P ( Z ℓ ) · P ( Z i | X i ) , i =1 n � � P ( y | x 1 , . . . , x n ) = P ( z ℓ ) · P ( z i | x i ) . z 1 ,..., z n , z ℓ : y = f ( z 1 ,..., z n , z ℓ ) i =1 4 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Possibilistic Networks with ICI Definition (Uncertain Function) π ( y | x 1 , . . . , x n ) = z 1 ,..., z n : y = f ( z 1 ,..., z n ) ∗ i =1 ,..., n π ( z i | x i ) , max � 1 if y = f ( x 1 , . . . x n ); π ( y | x 1 , . . . , x n ) = 0 otherwise. Leaky ICI Model π ( y | x 1 , . . . , x n ) = z 1 ,..., z n , z ℓ : y = f ( z 1 ,..., z n , z ℓ ) ∗ i =1 ,..., n π ( z i | x i ) ∗ π ( z ℓ ) . max 5 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates Uncertain OR Gate π ( Z i | X i ) x i ¬ x i z i 1 0 ¬ z i κ i 1 Variables are Boolean (i.e., Y = y or ¬ y , etc.). f ( Z 1 , . . . , Z n ) = � n i =1 Z i . Causes may fail to produce their effects Z i = ¬ z i ⇔ X i = x i did not cause Y = y due to some inhibitor. Then we must define π ( z i | x i ) = 1 and π ( ¬ z i | x i ) = κ i < 1. π ( z i | ¬ x i ) = 0, since when X i is absent, it does not cause y . 6 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates Conditional Table for the Uncertain OR Gate z 1 ,..., z n : z 1 ∨···∨ z n =1 ∗ n π ( y | X 1 , . . . , X n ) = max i =1 π ( z i | X i ) n = max i =1 π ( z i | X i ) ∗ ( ∗ j � = i max( π ( z j | X j ) π ( ¬ z j | X j )); z 1 ,..., z n : z 1 ∨···∨ z n =0 ∗ n π ( ¬ y | X 1 , . . . , X n ) = max i =1 π ( z i | X i ) = π ( ¬ z 1 | X 1 ) ∗ · · · ∗ π ( ¬ z n | X n ) . For n = 2 π ( y | X 1 X 2 ) x 1 ¬ x 1 π ( ¬ y | X 1 X 2 ) x 1 ¬ x 1 x 2 1 1 x 2 κ 1 ∗ κ 2 κ 2 ¬ x 2 1 0 ¬ x 2 1 κ 1 7 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain OR and AND Gates Uncertain AND Gate π ( Z i | X i ) x i ¬ x i z i 1 0 ¬ z i κ i 1 Same local conditional table as OR However, X i = x i is now a necessary cause for Y = y . f ( Z 1 , . . . , Z n ) = � n i =1 Z i . For n = 2 π ( y | X 1 X 2 ) x 1 ¬ x 1 π ( ¬ y | X 1 X 2 ) x 1 ¬ x 1 x 2 1 0 x 2 max( κ 1 , κ 2 ) 1 ¬ x 2 0 0 ¬ x 2 1 1 8 / 21

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 | X 1 X 2 ) ¬ x 1 π ( ¬ y | X 1 X 2 ) ¬ x 1 x 1 x 1 1 1 κ 1 ∗ κ 2 x 2 x 2 κ 2 ¬ x 2 1 ¬ x 2 1 κ ℓ κ 1 Leaky AND, for n = 2 π ( y | X 1 X 2 ) ¬ x 1 π ( ¬ y | X 1 X 2 ) ¬ x 1 x 1 x 1 1 max( κ 1 , κ 2 ) 1 x 2 κ L x 2 ¬ x 2 ¬ x 2 1 1 κ L κ L The 0 entries have been replaced by the leakage coefficient. 9 / 21

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 10 / 21

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. 11 / 21

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 Z i ) are valued on severity scale L = { 0 < 1 < · · · < m } Y = max( Z 1 , . . . , Z n ) z 1 ,..., z n : y =max( z 1 ,..., z n ) ∗ n π ( y | x 1 , . . . , x n ) = max i =1 π ( z i | x i ) n = max i =1 π ( Z i = y | x i ) ∗ ( ∗ j � = i Π( Z j ≤ y | x j )) . Possibility Table for for 3 levels of strength 0, 1, 2 π ( Z i | X i ) X i = 2 X i = 1 X i = 0 Z i = 2 1 0 0 κ 12 Z i = 1 1 0 i κ 02 κ 01 Z i = 0 1 i i 12 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain MAX and MIN Gates Uncertain MAX: Global Conditional Possibility Table n π ( Y = j | x ) = max i =1 π ( Z i = j | x i ) ∗ ( ∗ ℓ � = i Π( Z ℓ ≤ j | x ℓ )) For n = 2, m = 2, 3 levels of strength x π (2 | x ) π (1 | x ) π (0 | x ) max( κ 12 1 , κ 12 κ 02 1 ∗ κ 02 (2 , 2) 1 2 ) 2 κ 02 1 ∗ κ 01 (2 , 1) 1 1 2 κ 12 κ 02 (2 , 0) 1 1 1 κ 01 1 ∗ κ 02 (1 , 2) 1 1 2 κ 01 1 ∗ κ 01 (1 , 1) 0 1 2 κ 01 (1 , 0) 0 1 1 κ 12 κ 02 (0 , 2) 1 2 2 κ 01 (0 , 1) 0 1 2 (0 , 0) 0 0 1 13 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Uncertain MAX and MIN Gates Uncertain MIN z 1 ,..., z n : y =min( z 1 ,..., z n ) ∗ n π ( y | x 1 , . . . , x n ) = max i =1 π ( z i | x i ) n = max i =1 π ( Z i = y | x i ) ∗ ( ∗ j � = i Π( Z j ≥ y | x j )) . For n = 2, m = 2, 3 levels of strength x π (2 | x ) π (1 | x ) π (0 | x ) max( κ 12 1 , κ 12 max( κ 02 1 , κ 02 (2 , 2) 1 2 ) 2 ) max( κ 02 1 , κ 01 (2 , 1) 0 1 2 ) κ 12 (2 , 0) 0 1 1 max( κ 01 1 , κ 02 (1 , 2) 0 1 2 ) max( κ 01 1 , κ 01 (1 , 1) 0 1 2 ) (1 , 0) 0 0 1 κ 12 (0 , 2) 0 1 2 (0 , 1) 0 0 1 (0 , 0) 0 0 1 14 / 21

Introduction BNs with ICI Canonical Possibilistic Networks Implementation Application Conclusion Algorithm 1. uncertain-MAX ( Y , prm ) 1: π ( Y | X 1 , . . . , X n ) ← 0 2: for all x ∈ X 1 × . . . × X n do K ← { k : � cond i , k � ∈ prm , x | = cond i } { Select the 3: parameters that apply to x } for all y = ( y 1 , . . . , y � K � ) ∈ Y � K � do 4: β ← min i =1 ,..., � K � { κ iy i } 5: y ← max i =1 ,..., � K � { y i } ¯ 6: π (¯ y | x ) ← max { β, π (¯ y | x ) } 7: end for 8: 9: end for 10: return π ( Y | X 1 , . . . , X n ) 15 / 21

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 θ j ∈ { 1 , 2 , . . . } for each value y j of Y if at least θ j causes concur, Π( Y = y j | x ) = 1 16 / 21