Simulation methods for lower previsions Matthias C. M. Troffaes work partially supported by H2020 Marie Curie ITN, UTOPIAE, Grant Agreement No. 722734 Durham University, United Kingdom July, 2018 1
Outline Problem Description Imprecise Estimation Lower and Upper Estimators for the Minimum of a Function Bias of Lower and Upper Estimators Consistency of the Lower Estimator Discrepancy Bounds Confidence Interval from Lower and Upper Estimators Examples Toy Problem Two-Level Monte Carlo v1 Two-Level Monte Carlo v2 Importance Sampling Stochastic Approximation Kiefer-Wolfowitz Example 1 Example 2 Open Questions 2
Outline Problem Description Imprecise Estimation Lower and Upper Estimators for the Minimum of a Function Bias of Lower and Upper Estimators Consistency of the Lower Estimator Discrepancy Bounds Confidence Interval from Lower and Upper Estimators Examples Toy Problem Two-Level Monte Carlo v1 Two-Level Monte Carlo v2 Importance Sampling Stochastic Approximation Kiefer-Wolfowitz Example 1 Example 2 Open Questions 3
Problem Description Remember the natural extension of a gamble g : E ( g ) ≔ min p ∈M E p ( g ) (1) ◮ It represents the supremum buying price α you should be willing to pay for g ◮ We can use this natural extension for all statistical inference and decision making. ◮ how to evaluate the minimum in eq. (1) provided we have an estimator for E p ( g ) ? 4
Problem Description statistical inference under severe uncertainty lower simulation previsions importance imprecise optimization credal set sampling estimators 5
Outline Problem Description Imprecise Estimation Lower and Upper Estimators for the Minimum of a Function Bias of Lower and Upper Estimators Consistency of the Lower Estimator Discrepancy Bounds Confidence Interval from Lower and Upper Estimators Examples Toy Problem Two-Level Monte Carlo v1 Two-Level Monte Carlo v2 Importance Sampling Stochastic Approximation Kiefer-Wolfowitz Example 1 Example 2 Open Questions 6
Lower and Upper Estimators for the Minimum of a Function (see [12]) ◮ Ω = random variable, taking values in some subset of R k ◮ t = parameter taking values in some set T (assume T countable) ◮ θ ( t ) = arbitrary function of t ◮ ˆ θ Ω ( t ) = arbitrary estimator for θ : E (ˆ θ Ω ( t )) = θ ( t ) , (2) Aim Construct an estimator for the minimum of the function θ : θ ∗ ≔ inf t ∈T θ ( t ) . (3) Example Say for instance M = { p t : t ∈ T } , and let θ ( t ) ≔ E p t ( f ) . Then θ ∗ = E ( f ) . So estimation of θ ∗ = estimation of natural extension. 7
Lower and Upper Estimators for the Minimum of a Function Define the function ˆ τ Ω ∈ arg inf θ Ω ( t ) (4) t ∈T Theorem (Lower and Upper Estimator Theorem [12]) Assume Ω and Ω ′ are i.i.d. and let ˆ θ ∗ (Ω) ≔ ˆ ˆ θ Ω ( τ Ω ) = inf θ Ω ( t ) (5) t ∈T ˆ θ ∗ (Ω , Ω ′ ) ≔ ˆ θ Ω ( τ Ω ′ ) (6) Then θ ∗ (Ω) ≤ ˆ ˆ θ ∗ (Ω , Ω ′ ) (7) and E (ˆ θ ∗ (Ω)) ≤ θ ∗ ≤ E (ˆ θ ∗ (Ω , Ω ′ )) . (8) 8
Lower and Upper Estimators for the Minimum of a Function 0.55 0.50 θ ( t ) 0.45 ^ Ω ( t ) θ θ ^ Ω ' ( t ) θ 0.40 0.35 θ * 0.30 −0.5 0.0 0.5 t 9
Lower and Upper Estimators for the Minimum of a Function 0.55 0.50 θ ( t ) 0.45 ^ Ω ( t ) θ θ ^ Ω ' ( t ) θ 0.40 0.35 θ * 0.30 −0.5 0.0 0.5 t 10
Lower and Upper Estimators for the Minimum of a Function 0.55 0.50 θ ( t ) 0.45 ^ Ω ( t ) θ θ ^ Ω ' ( t ) θ 0.40 0.35 θ * 0.30 θ * ( Ω ) τ Ω −0.5 0.0 0.5 t 11
Lower and Upper Estimators for the Minimum of a Function 0.55 0.50 θ ( t ) 0.45 ^ Ω ( t ) θ θ ^ Ω ' ( t ) θ 0.40 0.35 θ * 0.30 θ * ( Ω ) τ Ω τ Ω ' −0.5 0.0 0.5 t 12
Lower and Upper Estimators for the Minimum of a Function 0.55 0.50 θ ( t ) 0.45 ^ Ω ( t ) θ θ ^ Ω ' ( t ) θ 0.40 θ *( Ω , Ω ') 0.35 θ * 0.30 θ * ( Ω ) τ Ω τ Ω ' −0.5 0.0 0.5 t 13
Bias of Lower and Upper Estimators ◮ ˆ θ ∗ (Ω) : used throughout the literature as an estimator for lower previsions not normally noted in the literature that it is negatively biased bias can be very large in general (even infinity)! ◮ ˆ θ ∗ (Ω , Ω ′ ) : introduced at last year’s WPMSIIP still cannot yet prove much about it it allows us to bound the bias without having to do hardcore stochastic process theory Theorem (Unbiased Case [12]) If there is a t ∗ ∈ T such that ˆ θ Ω ( t ∗ ) ≤ ˆ θ Ω ( t ) for all t ∈ T , then ˆ θ ∗ (Ω) = ˆ θ ∗ (Ω , Ω ′ ) = ˆ θ Ω ( t ∗ ) (9) and consequently, E (ˆ θ ∗ (Ω)) = θ ∗ = E (ˆ θ ∗ (Ω , Ω ′ )) . (10) (Condition not normally satisfied, but explains why it is a sensible choice.) 14
Consistency of the Lower Estimator Very often, an estimator may take the form of an empirical mean: n θ Ω , n ( t ) = 1 � ˆ ˆ θ V i ( t ) (11) n i = 1 where Ω ≔ ( V i ) i ∈ N and V i are i.i.d. Under mild conditions, this estimator is consistent: n →∞ P ( | ˆ θ Ω , n ( t ) − θ ( t ) | > ǫ ) = 0 lim (12) ◮ Under what conditions is ˆ θ ∗ n (Ω) a consistent estimator for θ ∗ , i.e. when do we have that n →∞ P ( | ˆ θ ∗ n (Ω) − θ ∗ | > ǫ ) = 0 lim (13) ◮ How large should n be? 15
Consistency of the Lower Estimator Simple case first: Theorem (Consistency: Finite Case [12]) If T is finite, then ˆ θ ∗ n (Ω) is a consistent estimator for θ ∗ . (Even though consistent, may require excessively large n to control bias!) General case, no positive answer in general, but consistency can be linked to a well-known condition in stochastic process theory: Theorem (Consistency: Sufficient Condition for General Case [12]) If the set of functions { ˆ θ ( · , t ): t ∈ T } is a Glivenko-Cantelli class, then ˆ θ ∗ n (Ω) is a consistent estimator for θ ∗ . 16
Discrepancy Bounds for the Lower Estimator Notation: Z n ( t ) ≔ ˆ θ Ω , n ( t ) − θ ( t ) (14) � d n ( s , t ) ≔ E (( Z n ( s ) − Z n ( t )) 2 ) (15) ∆ n ( A ) ≔ sup d n ( s , t ) (16) s , t ∈ A σ 2 t ∈T Var (ˆ t ∈T Var ( Z n ( t )) = inf θ Ω , n ( t )) n ≔ inf (17) Definition (Talagrand Functional) Define the Talagrand functional [10, p. 25] as: ∞ � 2 k / 2 ∆ n ( A k ( t )) γ 2 ( T , d n ) ≔ inf sup (18) A k t ∈ T k = 0 where the infimum is taken over all ‘admissible sequences of partitions of T ’. 17
Discrepancy Bounds for Empirical Mean Lower Estimator Theorem (Discrepancy Bounds for Empirical Mean Lower Estimator [12]) Assume ˆ θ ∗ n (Ω) ≔ 1 � n i = 1 ˆ θ V i ( t ) . There is a universal constant L > 0 such that, if ˆ θ Ω , n ( t ) is n sub-Gaussian, then � � ≤ L exp ( − nu 2 | ˆ θ ∗ n (Ω) − θ ∗ | > u ( σ 1 + γ 2 ( T , d 1 )) 2 ) P (19) and ≤ L σ 1 + γ 2 ( T , d 1 ) � � | ˆ E θ ∗ n (Ω) − θ ∗ | (20) √ n . Corollary (Consistency of Empirical Mean Lower Estimator [12]) If ˆ θ Ω , n ( t ) is sub-Gaussian, then ˆ θ ∗ n (Ω) is a consistent estimator for θ ∗ whenever the minimal standard deviation σ 1 and the Talagrand functional γ 2 ( T ′ , d 1 ) are finite. Issue: it is not easy to compute or to bound the Talagrand functional! 18
Empirical Mean Lower Estimator: How To Achieve Low Bias Inconsistency Example ◮ ˆ θ Ω , n ( t ) has non-zero variance across all t ◮ ˆ θ Ω , n ( s ) and ˆ θ Ω , n ( t ) are independent for all s � t ◮ T is infinite Then the Talagrand functional γ 2 ( T , d 1 ) is + ∞ . Important for 2-level Monte Carlo: don’t use i.i.d. samples in outer loop over t ∈ T ! Main Take-Home Message for Design of Estimators To get a low Talagrand functional (and hence a low bias), we want ˆ θ Ω , n ( s ) and ˆ θ Ω , n ( t ) to be as correlated as possible for all s � t . 19
Confidence Interval Theorem (Confidence Interval from Lower and Upper Estimators [12]) Let χ 1 , . . . , χ N , χ ′ 1 , . . . , χ ′ N be a sequence of i.i.d. realisations of Ω . Define Y ∗ ≔ (ˆ θ ∗ ( χ i )) N Y ∗ ≔ (ˆ θ ∗ ( χ i , χ ′ i )) N (21) i = 1 i = 1 Y ∗ be the sample means of these sequences, and let S ∗ and S ∗ be their sample Let ¯ Y ∗ and ¯ standard deviations. Let t N − 1 denote the usual two-sided critical value of the t-distribution with N − 1 degrees of freedom at confidence level 1 − α . Then, provided that sup x , t | ˆ θ ( x , t ) | < + ∞ , � � S ∗ S ∗ Y ∗ + t N − 1 ¯ , ¯ Y ∗ − t N − 1 √ √ (22) N N is an approximate confidence interval for θ ∗ with confidence level (at least) 1 − α . Why is this rather slow? Note: we can cheat and use ˆ θ ∗ ( χ ′ i , χ i ) instead for Y ∗ . Y ∗ with probability ≃ 1). This trick halves computational time (caveat: need ¯ Y ∗ ≤ ¯ 20
Outline Problem Description Imprecise Estimation Lower and Upper Estimators for the Minimum of a Function Bias of Lower and Upper Estimators Consistency of the Lower Estimator Discrepancy Bounds Confidence Interval from Lower and Upper Estimators Examples Toy Problem Two-Level Monte Carlo v1 Two-Level Monte Carlo v2 Importance Sampling Stochastic Approximation Kiefer-Wolfowitz Example 1 Example 2 Open Questions 21
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