Optimal Uncertainty Quantification of Quantiles

Transcription

1 Optimal Uncertainty Quantification of Quantiles Application to Industrial Risk Management. Merlin Keller 1, Jérôme Stenger 12 1 EDF R&D, France 2 Institut Mathématique de Toulouse, France ETICS 2018, Roscoff 1 / 12

2 1 Introduction 2 Robust Bounds on Quantiles 2 / 12

3 Optimal Uncertainty Quantification (OUQ) : [Owhadi et al., 2013] Principle Find optimal bounds for a quantity of interest Q(µ ), functional of an uncertain probability measure µ, known only to lie in some subset A of M 1(X ) : with : Q(A) = inf µ A Q(µ) Q(A) = sup µ A Q(µ) Q(A) Q(µ ) Q(A), A = {µ M 1(X ) Φ j (µ) c j, j = 1,..., N} the admissible subset, Distributionally Robust : Useful in risk-adverse situations when parametric assumptions are hard to justify Many applications in Industrial Risk Management (find two : exercise 1) 3 / 12

4 Optimal Uncertainty Quantification (OUQ) : [Owhadi et al., 2013] Principle Find optimal bounds for a quantity of interest Q(µ ), functional of an uncertain probability measure µ, known only to lie in some subset A of M 1(X ) : with : Q(A) = inf µ A Q(µ) Q(A) = sup µ A Q(µ) Q(A) Q(µ ) Q(A), A = {µ M 1(X ) Φ j (µ) c j, j = 1,..., N} the admissible subset, Distributionally Robust : Useful in risk-adverse situations when parametric assumptions are hard to justify Many applications in Industrial Risk Management (find two : exercise 1) Dyke Conception / Reliability Assessment 3 / 12

5 Optimal Uncertainty Quantification (OUQ) : [Owhadi et al., 2013] Principle Find optimal bounds for a quantity of interest Q(µ ), functional of an uncertain probability measure µ, known only to lie in some subset A of M 1(X ) : with : Q(A) = inf µ A Q(µ) Q(A) = sup µ A Q(µ) Q(A) Q(µ ) Q(A), A = {µ M 1(X ) Φ j (µ) c j, j = 1,..., N} the admissible subset, Distributionally Robust : Useful in risk-adverse situations when parametric assumptions are hard to justify Many applications in Industrial Risk Management (find two : exercise 1) Dyke Conception / Reliability Assessment : Q(µ) := P µ[z h > 0], Z µ water level, h dyke height, Z h overflow 3 / 12

6 Optimal Uncertainty Quantification (OUQ) : [Owhadi et al., 2013] Principle Find optimal bounds for a quantity of interest Q(µ ), functional of an uncertain probability measure µ, known only to lie in some subset A of M 1(X ) : with : Q(A) = inf µ A Q(µ) Q(A) = sup µ A Q(µ) Q(A) Q(µ ) Q(A), A = {µ M 1(X ) Φ j (µ) c j, j = 1,..., N} the admissible subset, Distributionally Robust : Useful in risk-adverse situations when parametric assumptions are hard to justify Many applications in Industrial Risk Management (find two : exercise 1) Dyke Conception / Reliability Assessment : Q(µ) := P µ[z h > 0], Z µ water level, h dyke height, Z h overflow Nuclear Accident Risk Assessment 3 / 12

7 Optimal Uncertainty Quantification (OUQ) : [Owhadi et al., 2013] Principle Find optimal bounds for a quantity of interest Q(µ ), functional of an uncertain probability measure µ, known only to lie in some subset A of M 1(X ) : with : Q(A) = inf µ A Q(µ) Q(A) = sup µ A Q(µ) Q(A) Q(µ ) Q(A), A = {µ M 1(X ) Φ j (µ) c j, j = 1,..., N} the admissible subset, Distributionally Robust : Useful in risk-adverse situations when parametric assumptions are hard to justify Many applications in Industrial Risk Management (find two : exercise 1) Dyke Conception / Reliability Assessment : Q(µ) := P µ[z h > 0], Z µ water level, h dyke height, Z h overflow Nuclear Accident Risk Assessment : Q(µ) := sup{t > 0 P µ[t t] p}, T µ maximal temperature inside nuclear reactor, p safety requirement 3 / 12

8 Optimal Uncertainty Quantification (OUQ) : [Owhadi et al., 2013] Principle Find optimal bounds for a quantity of interest Q(µ ), functional of an uncertain probability measure µ, known only to lie in some subset A of M 1(X ) : with : Q(A) = inf µ A Q(µ) Q(A) = sup µ A Q(µ) Q(A) Q(µ ) Q(A), A = {µ M 1(X ) Φ j (µ) c j, j = 1,..., N} the admissible subset, Distributionally Robust : Useful in risk-adverse situations when parametric assumptions are hard to justify Many applications in Industrial Risk Management (find two : exercise 1) Dyke Conception / Reliability Assessment : Q(µ) := P µ[z h > 0], Z µ water level, h dyke height, Z h overflow Nuclear Accident Risk Assessment : Q(µ) := sup{t > 0 P µ[t t] p}, T µ maximal temperature inside nuclear reactor, p safety requirement and also : Probabilistic Seismic Hazard Assessment (PSHA), Extreme weather forecasting, Environmental risk assessment, Ecotoxicology,... 3 / 12

9 Robust Bayesian Inference (RBI) : [Rios Insua and Ruggeri, 2000] RBI as a special case of OUQ µ prior distribution on parameter θ in statistical model Y f θ dλ, belonging to a class A of admissible priors P Y (µ) posterior distribution of θ given Y, according to Bayes theorem : dp Y (µ)(θ) = f θ (Y )dµ(θ) ν fν(y )dµ(ν) (1) Derive optimal bounds on interest quantity of posterior distribution replace Q(µ) by Q(P Y (µ)) (or Q by Q P Y ) 4 / 12

10 Robust Bayesian Inference (RBI) : [Rios Insua and Ruggeri, 2000] RBI as a special case of OUQ µ prior distribution on parameter θ in statistical model Y f θ dλ, belonging to a class A of admissible priors P Y (µ) posterior distribution of θ given Y, according to Bayes theorem : dp Y (µ)(θ) = f θ (Y )dµ(θ) ν fν(y )dµ(ν) (1) Derive optimal bounds on interest quantity of posterior distribution replace Q(µ) by Q(P Y (µ)) (or Q by Q P Y ) OUQ as a special case of RBI OUQ corresponds to the no-data case (f θ := f does not depend on θ) Proof : (1) reduces to P Y (µ), whence Q(P Y (µ)) reduces to Q(µ) both formulations are equivalent, and special cases of the Dempster Schaeffer Theory (DST) (M. Couplet, private conversation) 4 / 12

11 Main result Theorem (Measure affine functionals over generalized moment classes) If : Q(µ) is measure affine (e.g. Q(µ) := E µ[q], q bounded above or below) A = {µ M 1(X ) E µ[ϕ j ] c j, j = 1,..., N} for measurable functions ϕ j A = {µ A µ = N 0=1 w iδ xi } extremal admissible probability measures Then : Q(A) = Q(A ) ; Q(A) = Q(A ) 5 / 12

12 Main result Theorem (Measure affine functionals over generalized moment classes) If : Q(µ) is measure affine (e.g. Q(µ) := E µ[q], q bounded above or below) A = {µ M 1(X ) E µ[ϕ j ] c j, j = 1,..., N} for measurable functions ϕ j A = {µ A µ = N 0=1 w iδ xi } extremal admissible probability measures Then : Q(A) = Q(A ) ; Q(A) = Q(A ) Implementation To find Q(A) (resp.q(a)) : Minimize (resp. Maximize) Q(µ) = N i=0 w iq(x i ) wrt : (w i, x i ) 0 i N subject to : N i=0 w iϕ j (x i ) c j, for j = 1,..., N Constrained optimization problem, solvabe by (almost) off-the shelf methods if q, ϕ j analytical functions : Mystic framework [McKerns et al., 2012] 5 / 12

13 1 Introduction 2 Robust Bounds on Quantiles 6 / 12

14 A Simple Result Theorem (Quantile-CDF duality) Assume X = R + Let : Then : F µ(x) = P µ[x x] pointwise cdf evaluation functional Q µ(p) = inf{x > 0 F µ(x) p} order-p quantile Q A(p) = inf{x > 0 F A(x) p} 7 / 12

15 A Simple Result Theorem (Quantile-CDF duality) Assume X = R + Let : Then : Proof. F µ(x) = P µ[x x] pointwise cdf evaluation functional Q µ(p) = inf{x > 0 F µ(x) p} order-p quantile Q A(p) = inf{x > 0 F A(x) p} µ A : {x > 0 F A(x) p} {x > 0 F µ(x) p} inf{x > 0 F A(x) p} Q µ(p) inf{x > 0 F A(x) p} Q A(p). Assuming a strict inequality, x 0, s.t. : x 0 < inf{x > 0 F A (x) p} F A (x 0 ) < p µ, F µ(x 0 ) < p Q A (p) < x 0 Q µ(p) < x 0 F µ(x 0 ) p, leading to a contradiction 7 / 12

16 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) 8 / 12

17 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) 8 / 12

18 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) 8 / 12

19 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) 8 / 12

20 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) 8 / 12

21 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) Challenges 8 / 12

22 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) Challenges How to choose x t (a t 1, b t 1)? 8 / 12

23 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) Challenges How to choose x t (a t 1, b t 1)? Can we guarantee Q A(p) is nontrivial (< )? 8 / 12

24 Application : Inversion of CDF Bounds Sequential construction of Quantile Upper Bound Initialization (t = 0) : (a t, b t) = (0, + ) For t = 1, 2,... : Choose x t (a t 1, b t 1 ) Calculate F A (x t) If F A (x t) p, set a t := x t If F A (x t) > p, set b t := x t Stop when b t a t < ε b t is then an ε-approximation of Q A(p) Challenges How to choose x t (a t 1, b t 1)? Can we guarantee Q A(p) is nontrivial (< )? What if Q(µ) is an extreme quantile on Y = G(X ), with X µ and G a costly computer model? Can we develop an EGO-like approach? 8 / 12

25 Alternative : Direct quantile optimization Q µ(p) not a measure affine functional However, quantile-cdf duality ensures that main result still applies : OUQ for quantiles To find Q A(p) (resp.q A(p)) : Minimize (resp. Maximize) Q µ(p) = x (i ) wrt : (w i, x i ) 0 i N where : i = min 0 i N il=0 w (i) p x (0)... x (N) w i > 0 and N i=1 w i = 1 subject to : N i=0 w iϕ j (x i ) c j, for j = 1,..., N 9 / 12

26 Alternative : Direct quantile optimization Q µ(p) not a measure affine functional However, quantile-cdf duality ensures that main result still applies : OUQ for quantiles To find Q A(p) (resp.q A(p)) : Minimize (resp. Maximize) Q µ(p) = x (i ) wrt : (w i, x i ) 0 i N where : i = min 0 i N il=0 w (i) p x (0)... x (N) w i > 0 and N i=1 w i = 1 subject to : N i=0 w iϕ j (x i ) c j, for j = 1,..., N Main difficulty Objective function Q p(µ) = Q p((w i, x i ) 0 i N ) irregular and non convex 9 / 12

27 Case study : Quantile of nonlinear transform under product measure Problem specification (simplified version) Q µ(p) = sup{y > 0, P µ[g(x ) y] p}, where : X = (X 1,..., X d ) µ over [0, 1] d G : [0, 1] d R + potentially costly A = {µ = d k=1µ k E µk [X k ] = m k, 1 k d} Extreme set of product measures [Owhadi et al., 2013] show that Q(A) = Q(A ) where : Q(µ) measure affine (extendable to quantiles by duality with CDF) ( ) A = {µ = d k=1 wk δ xk,0 + (1 w k )δ xk,1 wk x k,0 + (1 w k )x k,1 = m k } 10 / 12

28 To be continued... THANKS FOR YOUR ATTENTION!! 11 / 12

29 References McKerns, M., Owhadi, H., Scovel, C., Sullivan, T. J., and Ortiz, M. (2012). The optimal uncertainty algorithm in the mystic framework. CoRR, abs/ Owhadi, H., Scovel, C., Sullivan, T. J., McKerns, M., and Ortiz, M. (2013). Optimal Uncertainty Quantification. SIAM Rev., 2(55) : Rios Insua, D. and Ruggeri, F. (2000). Robust Bayesian Analysis. Springer-Verlag. 12 / 12