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 t ( f ) 2

Estimators for lower expectations E P 1 ( f ) E P 2 ( f ) 3

Estimators for lower expectations E P 1 ( f ) E P 2 ( f ) E P 2 ( f ) 3

Estimators for lower expectations Infinite set of probability measures test 4

Estimators for lower expectations 1. Fix sampling P 5

Estimators for lower expectations 1. Fix sampling P 2. Find f t such that E P ( f t ) = E P t ( f ) 5

Estimators for lower expectations 1. Fix sampling P 2. Find f t such that E P ( f t ) = E P t ( f ) → Importance Sampling 5

Estimators for lower expectations 1. Fix sampling P 2. Find f t such that E P ( f t ) = E P t ( f ) → Importance Sampling n ? = ˆ E P ( f ) “ X P ” ≈ inf X f t E k t k =1 5

Bias negative unbiased positive E(ˆ E( f ) E) E(ˆ E) = E( f ) E(ˆ E) E( f ) 6

Bias negative unbiased positive E(ˆ E( f ) E) E(ˆ E) = E( f ) E(ˆ E) E( f ) 6

Bias negative unbiased positive E(ˆ E( f ) E) E(ˆ E) = E( f ) E(ˆ E) E( f ) can only get closer constant it depends E E E n n n 6

Bias negative unbiased positive E(ˆ E( f ) E) E(ˆ E) = E( f ) E(ˆ E) E( f ) can only get closer constant it depends E E E n n n 6

consistency 7

consistency 1. In probability „˛ ˛ « ˛ ˆ n →∞ P ∞ lim E n − E( f ) = 0 ˛ ˛ > › ˛ ˛ ˛ 2. Almost surely „ « n →∞ ˆ P ∞ lim E n = E( f ) = 0 7

consistency 1. In probability „˛ ˛ « ˛ ˆ n →∞ P ∞ lim E n − E( f ) = 0 ˛ ˛ > › ˛ ˛ ˛ 2. Almost surely „ « n →∞ ˆ P ∞ lim E n = E( f ) = 0 7

Consistency n t E P t ( f ) = E P ( f ) ? “ X P ” → inf as n → ∞ inf X f t k t k =0 8

Consistency n t E P t ( f ) = E P ( f ) ? “ X P ” → inf as n → ∞ inf X f t k t k =0 ⇑ ˛ ˛ n ˛ ˛ − E P t ( f ) ? “ X P ” sup X → 0 as n → ∞ ˛ f t ˛ ˛ k ˛ ˛ ˛ t k =0 ˛ ˛ 8

Consistency When is this the case? 9

Consistency When is this the case? • restrictions on size of T 9

Consistency When is this the case? • restrictions on size of T • continuity conditions for f t 9

Consistency When is this the case? • restrictions on size of T • continuity conditions for f t R n ⊃ T compact R n ⊃ T bounded for all › > 0 : T has a finite › -cover p t ( x ) cont. diff. in ( x; t ) finite T �∇ t p t ( x ) � < F ( x ) | p t ( x ) − p s ( x ) | 6 d ( s; t ) F ( x ) E P (sup t ∈ T p t ) < + ∞ easier but more restrictive more general but more complex 9

Practical Example q P ( g ( X ) 6 0) = 1 − E P “ ” I { g ( X ) > 0 } X L Fetz, T., & Oberguggenberger, M. (2016). Imprecise random variables, random sets, and Monte Carlo simulation. 10

See you at my poster. 11