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.

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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 vessels X1 = Type of other vessel X2 = Proximity of other vessel X3 = Wind speed X4 = Wind direction X5 = Current speed X6 = Current direction X7 = Visibility

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P(Event | X1,…,Xn)

Event = Incoming vessel contains RDD X1 = Last country docked X2 = 2nd to last country docked X3 = 3rd to last country docked X4 = Frequency of US calls X5 = Vessel ownership X6 = Type of vessel X7 = Type of crew

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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

What is the probability

• f a collision?

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

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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

• 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

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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) = Xi

Tβ + ui, j

= p0 exp(RTβ) p0 exp(L

Tβ) = 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 9

Likelihood of Collision

yi, j = ln(zi, j) = Xi

Tβ + ui, j

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u = u1  up ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ~ MVNormal 0,Σ

( )

θ ui = µi −θ

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π(θ;µ,Σ) ∝ exp − θ − µ*

( )

2 / 2σ * 2

( )

µ* = 1

TΣ−1µ

1

TΣ−11

σ *2 = 1 1

TΣ−11

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u = u1  up ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ~ MVNormal 0,Σ

( )

θ ui = µi −θ Xi

Tβ

ui, j = yi, j − Xi

Tβ

ui = ui,1  ui,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 − 12tr VΣ−1

( )

{ } exp − 12tr

 B − β1

T

( )

T XTX 

B − β1

T

( )Σ−1

( )

{ }

Σ*

β =

X T X

( )

−1

1

TΣ−11

µ*

β =

ˆ BΣ−11 1

TΣ−11

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Σ

( ) ~ Inv −Wishart G,m ( )

β | Y,X,Σ

( ) ~ MVNormal ϕ,

A 1

TΣ−11

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Σ | Y,X

( ) ~ Inv −Wishart G + V,m + N ( )

β | Y,X,Σ

( ) ~ MVNormal

A−1 + XTX

( )

−1 XTX

ˆ BΣ−11 1

TΣ−11 + A−1ϕ

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , A−1 + XTX

( )

−1

1

TΣ−11

⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟

V = Y − X ˆ B

( )

T Y − X ˆ

B

( )

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

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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!

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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 = 0.73 Independent [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,...,z p

( )

r

1,...,rp

( )

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