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Monte Carlo Estimation for Imprecise Probabilities Basic Properties
Arne Decadt Gert de Cooman Jasper De Bock
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Setting
Monte Carlo
1 n
n X k=1
f (XP
k ) ≈ EP(f )
Imprecise Probability
P = {Pt : t ∈ T} EP(f ) = inf
t EPt(f )
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Monte Carlo Estimation for Imprecise Probabilities Basic Properties - - PowerPoint PPT Presentation
Monte Carlo Estimation for Imprecise Probabilities Basic Properties - - PowerPoint PPT Presentation
Monte Carlo Estimation for Imprecise Probabilities Basic Properties Arne Decadt Gert de Cooman Jasper De Bock 1 Setting Monte Carlo 1 n f ( X P k ) E P ( f ) X n k =1 Imprecise Probability P = { P t : t T } E P ( f ) = inf t E P
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SLIDE 3
Estimators for lower expectations
EP1 (f ) EP2 (f )
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Estimators for lower expectations
EP1 (f ) EP2 (f ) EP2 (f )
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Estimators for lower expectations
Infinite set of probability measures test
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Estimators for lower expectations
- 1. Fix sampling P
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Estimators for lower expectations
- 1. Fix sampling P
- 2. Find ft such that EP (ft) = EPt (f )
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Estimators for lower expectations
- 1. Fix sampling P
- 2. Find ft such that EP (ft) = EPt (f )
→ Importance Sampling
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Estimators for lower expectations
- 1. Fix sampling P
- 2. Find ft such that EP (ft) = EPt (f )
→ Importance Sampling EP(f )
?
≈ inf
t n X k=1
ft
“
XP
k ”
= ˆ E
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Bias
negative unbiased positive
E(ˆ E) E(f ) E(ˆ E) = E(f ) E(f ) E(ˆ E)
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Bias
negative unbiased positive
E(ˆ E) E(f ) E(ˆ E) = E(f ) E(f ) E(ˆ E)
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Bias
negative unbiased positive
E(ˆ E) E(f ) E(ˆ E) = E(f ) E(f ) E(ˆ E)
can only get closer constant it depends
E
n
E
n
E
n
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SLIDE 13
Bias
negative unbiased positive
E(ˆ E) E(f ) E(ˆ E) = E(f ) E(f ) E(ˆ E)
can only get closer constant it depends
E
n
E
n
E
n
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SLIDE 14
consistency
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consistency
- 1. In probability
lim
n→∞ P ∞ „˛ ˛ ˛ ˛ˆ
En − E(f )
˛ ˛ ˛ ˛ > › «
= 0
- 2. Almost surely
P ∞
„
lim
n→∞ ˆ
En = E(f )
«
= 0
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SLIDE 16
consistency
- 1. In probability
lim
n→∞ P ∞ „˛ ˛ ˛ ˛ˆ
En − E(f )
˛ ˛ ˛ ˛ > › «
= 0
- 2. Almost surely
P ∞
„
lim
n→∞ ˆ
En = E(f )
«
= 0
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SLIDE 17
Consistency
inf
t n X k=0
ft
“
XP
k ” ?
→ inf
t EPt (f ) = EP(f )
as n → ∞
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Consistency
inf
t n X k=0
ft
“
XP
k ” ?
→ inf
t EPt (f ) = EP(f )
as n → ∞
⇑ sup
t ˛ ˛ ˛ ˛ ˛ ˛ n X k=0
ft
“
XP
k ”
− EPt (f )
˛ ˛ ˛ ˛ ˛ ˛ ?
→ 0
as n → ∞
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Consistency
When is this the case?
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Consistency
When is this the case?
- restrictions on size of T
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SLIDE 21
Consistency
When is this the case?
- restrictions on size of T
- continuity conditions for ft
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SLIDE 22
Consistency
When is this the case?
- restrictions on size of T
- continuity conditions for ft
finite T Rn ⊃ T compact pt(x) cont. diff. in (x; t) EP(supt∈T pt) < +∞ Rn ⊃ T bounded ∇tpt(x) < F(x) for all › > 0 : T has a finite ›-cover |pt(x) − ps(x)| 6 d(s; t)F(x) easier but more restrictive more general but more complex 9
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Practical Example
q X L
P(g(X) 6 0) = 1 − EP “ I{g(X)>0}
” Fetz, T., & Oberguggenberger, M. (2016). Imprecise random variables, random sets, and Monte Carlo simulation.
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SLIDE 24
See you at my poster.
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