overlapping expert information: learning about...
TRANSCRIPT
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.
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Extending Winkler’s Multivariate Normal
Aggregation
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P(Event | X1,…,Xn)
P(Event)
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P(Event | X1,…,Xn)
Event = Collision between two vesselsX1 = Type of other vesselX2 = Proximity of other vesselX3 = Wind speedX4 = Wind directionX5 = Current speedX6 = Current directionX7 = Visibility
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P(Event | X1,…,Xn)
Event = Incoming vessel contains RDDX1 = Last country dockedX2 = 2nd to last country dockedX3 = 3rd to last country dockedX4 = Frequency of US callsX5 = Vessel ownershipX6 = Type of vesselX7 = Type of crew
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Issaquah class ferry On the Bremerton to Seattle route
Crossing situation within 15 minutesOther vessel is a navy vessel
No other vessels aroundGood visibility
Negligible wind
What is the probability of a collision?
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Issaquah class ferry On the Bremerton to Seattle route
Crossing situation within 15 minutesOther vessel is a navy vessel
No other vessels aroundGood visibility
Negligible wind
Issaquah class ferry On the Bremerton to Seattle route
Crossing situation within 15 minutesOther vessel is a product tanker
No other vessels aroundGood visibility
Negligible wind
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Issaquah Ferry Class -SEA-BRE(A) Ferry Route -
Navy 1st Interacting Vessel Product TankerCrossing 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 5 4 3 2 1 2 3 4 5 6 7 8 99 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 99 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9
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P(Event | X , p0 ,β ) = p0 exp X Tβ( )
P(Event | R,β )P(Event | L,β )
yi, j = ln(zi, j ) = XiTβ + ui, j
=
p0 exp(RTβ )p0 exp(LTβ )
= exp (R − L)T β( )
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9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9Likelihood of Collision
yi, j = ln(zi, j ) = XiTβ + ui, j
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u =u1up
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟~ MVNormal 0,Σ( )
θ
ui = µi −θ
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π (θ;µ,Σ)∝ exp − θ − µ*( )2 / 2σ *2( )
µ* = 1TΣ−1µ1TΣ−11
σ *2 = 11TΣ−11
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u =u1up
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟~ MVNormal 0,Σ( )
θ
ui = µi −θ
XiTβ
ui, j = yi, j − XiTβ
ui =ui,1ui,p
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟~ MVNormal 0,Σ( )
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y1,1 y1,p yN ,1 yN ,p
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
x1,1 x1,q xN ,1 xN ,q
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
β1 β1 βq βq
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟+
u1,1 u1,p uN ,1 uN ,p
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Y = Xβ1T +U
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p Y |X,β,Σ( )∝ Σ −N
2 exp − 12 tr VΣ
−1( ){ } exp − 12 tr
B− β1T( )T XTX
B− β1T( )Σ−1( ){ }
Σ*
β =X T X( )−1
1TΣ−11 µ*
β = B̂Σ−111TΣ−11
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Σ( ) ~ Inv −Wishart G,m( )
β |Y,X,Σ( ) ~ MVNormal ϕ, A1TΣ−11
⎛⎝⎜
⎞⎠⎟
Σ |Y,X( ) ~ Inv −Wishart G +V,m + N( )
β |Y,X,Σ( ) ~ MVNormal A−1 +XTX( )−1 XTX B̂Σ−11
1TΣ−11+A−1ϕ
⎛⎝⎜
⎞⎠⎟,A−1 +XTX( )−11TΣ−11
⎛
⎝⎜⎜
⎞
⎠⎟⎟
V = Y −XB̂( )T Y −XB̂( )
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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
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Assume independence between the experts a priori
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Comparing the two
scenarios we pictured earlier a priori
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Analysis with dependenceDoesn’t dependence
between experts increase posterior variance?
Analysis with independence
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Experts 1, 3 and 7 are correlatedExperts 2, 4 and 6 are correlatedExperts 5 and 8 are negatively or uncorrelated with other experts
Remember we assumed independence a priori, but we learnt
about Σ!
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Comparing the two
scenarios we pictured earlier Text
Prior [1.88*10-35, 5.32*1034]Dependent [4.38,5.84] ½ width = 0.73Independent [4.43,7.04] ½ width = 1.3
Text
90% Credibility
Interval
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Getting the Right Mix of Experts
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z1,..., zp( )
r1,...,rp( )
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