Overlapping Expert Information: Learning about Dependencies in - PowerPoint PPT Presentation

Overlapping Expert Information: Learning about Dependencies in Expert Judgment Jason R. W. Merrick Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 1

Extending Winkler’s Multivariate Normal Aggregation 2

P(Event) P(Event | X 1 ,…,X n ) 3

P(Event | X 1 ,…,X n ) Event = Collision between two vessels X 1 = Type of other vessel X 2 = Proximity of other vessel X 3 = Wind speed X 4 = Wind direction X 5 = Current speed X 6 = Current direction X 7 = Visibility 4

P(Event | X 1 ,…,X n ) Event = Incoming vessel contains RDD X 1 = Last country docked X 2 = 2nd to last country docked X 3 = 3rd to last country docked X 4 = Frequency of US calls X 5 = Vessel ownership X 6 = Type of vessel X 7 = Type of crew 5

What is the probability of a collision? Issaquah class ferry On the Bremerton to Seattle route Crossing situation within 15 minutes Other vessel is a navy vessel No other vessels around Good visibility Negligible wind 6

Issaquah class ferry Issaquah class ferry On the Bremerton to Seattle route On the Bremerton to Seattle route Crossing situation within 15 minutes Crossing situation within 15 minutes Other vessel is a navy vessel Other vessel is a product tanker No other vessels around No other vessels around Good visibility Good visibility Negligible wind Negligible wind 7

Issaquah Ferry Class - SEA-BRE(A) Ferry Route - Navy 1st Interacting Vessel Product Crossing Traffic Scenario 1st Vessel - < 1 mile Traffic Proximity 1st Vessel - No Vessel 2nd Interacting Vessel - No Vessel Traffic Scenario 2nd Vessel - No Vessel Traffic Proximity 2nd Vessel - > 0.5 Miles Visibility - Along Ferry Wind Direction - 0 Wind Speed - Likelihood of Collision - 9 8 7 6 7 6 5 4 3 2 1 2 3 4 5 6 6 7 8 9 8

( ) P ( Event | X , p 0 , β ) = p 0 exp X T β = p 0 exp( R T β ) P ( Event | R , β ) ( ) T β ) = exp ( R − L ) T β P ( Event | L , β ) p 0 exp( L y i , j = ln( z i , j ) = X i T β + u i , j 9

y i , j = ln( z i , j ) = X i T β + u i , j 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 Likelihood of Collision 10

θ u i = µ i − θ ⎛ ⎞ u 1 ⎜ ⎟ ( ) u = ~ MVNormal 0, Σ  ⎜ ⎟ ⎜ ⎟ u p ⎝ ⎠ 11

( ) ( ) 2 / 2 σ * π ( θ ; µ , Σ ) ∝ exp − θ − µ * 2 T Σ − 1 µ 1 σ *2 = µ * = 1 T Σ − 1 1 T Σ − 1 1 1 1 12

θ T β X i u i = µ i − θ u i , j = y i , j − X i T β ⎛ ⎞ ⎛ ⎞ u 1 u i ,1 ⎜ ⎟ ⎜ ⎟ ( ) ( ) u = ~ MVNormal 0, Σ  u i = ~ MVNormal 0, Σ ⎜ ⎟  ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ u p ⎝ ⎠ u i , p ⎝ ⎠ 13

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞   β 1 β 1   y 1,1 y 1, p x 1,1 x 1, q u 1,1 u 1, p ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = +             ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ β q β q ⎟     y N ,1 y N , p x N ,1 x N , q u N ,1 u N , p ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Y = X β 1 T + U 14

{ } ( ) { } exp − 12 tr  T X T X  ( ) ( ) Σ − 1 ( ) − N ( ) ∝ Σ p Y | X , β , Σ 2 exp − 12 tr V Σ − 1 B − β 1 B − β 1 T T ( ) ˆ − 1 X T X B Σ − 1 1 β = β = µ * Σ * T Σ − 1 1 1 T Σ − 1 1 1 15

( ) ~ Inv − Wishart G , m ( ) Σ ⎛ ⎞ A ( ) ~ MVNormal ϕ , β | Y , X , Σ ⎜ ⎟ ⎝ T Σ − 1 1 ⎠ 1 ( ) ( ) T Y − X ˆ V = Y − X ˆ B B ( ) ~ Inv − Wishart G + V , m + N ( ) Σ | Y , X ( ) ⎟ , A − 1 + X T X ⎛ ⎞ − 1 ⎛ ⎞ B Σ − 1 1 ˆ ( ) A − 1 + X T X − 1 X T X ( ) ~ MVNormal β | Y , X , Σ ⎜ T Σ − 1 1 + A − 1 ϕ ⎟ ⎜ T Σ − 1 1 ⎜ ⎟ ⎝ ⎠ 1 1 ⎝ ⎠ 16

Description Notation Values Ferry route and class FR_FC 26 Type of 1st interacting vessel TT_1 13 Scenario of 1st interacting vessel TS_1 4 Proximity of 1st interacting vessel TP_1 Binary Type of 2nd interacting vessel TT_2 5 Scenario of 2nd interacting vessel TS_2 4 Proximity of 2nd interacting vessel TP_2 Binary Visibility VIS Binary Wind direction WD Binary Wind speed WS Continuous 17

Assume independence between the experts a priori 18

Comparing the two scenarios we pictured earlier a priori 19

Doesn’t Analysis with dependence dependence between experts increase posterior variance? Analysis with independence 20

Experts 1, 3 and 7 are correlated Experts 2, 4 and 6 are correlated Experts 5 and 8 are negatively or uncorrelated with other experts Remember we assumed independence a priori, but we learnt about Σ ! 21

Comparing the two scenarios we pictured Text Text earlier 90% Prior [1.88*10 -35 , 5.32*10 34 ] Credibility Dependent [4.38,5.84] ½ width = 0.73 Interval Independent [4.43,7.04] ½ width = 1.3 22

Getting the Right Mix of Experts 23

( ) z 1 ,..., z p ( ) r 1 ,..., r p 24

25

26

27

28

29

30

31

32

33

34

35

36

37