The Geometry of Hypothesis Testing over Convex Cones

Transcription

1 The Geometry of Hypothesis Testing over Convex Cones Yuting Wei Department of Statistics, UC Berkeley BIRS workshop on Shape-Constrained Methods Jan 30rd, 2018 joint work with: Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

2 Cone testing problem Observation model: y = θ + σw, w N(0, I d ) Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

3 Cone testing problem Observation model: y = θ + σw, w N(0, I d ) Inference problem: H 0 : θ C 0 v.s. H 1 : θ C 1 \C 0 Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

4 Cone testing problem Observation model: y = θ + σw, w N(0, I d ) Inference problem: H 0 : θ C 0 v.s. H 1 : θ C 1 \C 0 C 0 C 1 closed, convex cones ( x C then ax C, a 0) Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

5 Applications In the fields of detection of treatment effects signal detection in radar processing trend detection in econometrics Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

6 Example: Treatment effect detection average treatment effect of d = 10 different dosages of a drug non-negative and un-ordered space unknown average effect Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

7 Example: Treatment effect detection average treatment effect of d = 10 different dosages of a drug non-negative and un-ordered space unknown average effect observed outcomes Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

8 Example: Treatment effect detection average treatment effect of d = 10 different dosages of a drug non-negative and un-ordered space unknown average effect observed outcomes Inference Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

9 Example: Treatment effect detection average treatment effect of d = 10 different dosages of a drug monotonic space unknown average effect Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

10 Example: Treatment effect detection average treatment effect of d = 10 different dosages of a drug monotonic space unknown average effect observed outcomes Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

11 Example: Treatment effect detection average treatment effect of d = 10 different dosages of a drug monotonic space unknown average effect observed outcomes Inference Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

12 Applications In the fields of detection of treatment effects signal detection in radar processing trend detection in econometrics Bühlmann 03, Meinshausen 03, Meyer 03, Sen and Meyer 17 Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

13 Applications In the fields of detection of treatment effects signal detection in radar processing trend detection in econometrics More statistical models linear regression y = X β + w : = θ + w C 0 : = {0} v.s. C 1 : = range(x ) fitness of a linear model: y = f (x n 1 ) + w : = θ + w, C 0 : = {X β β R k } v.s. C 1 : = convex cone Bühlmann 03, Meinshausen 03, Meyer 03, Sen and Meyer 17 Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

14 Overview Q: How to solve these constrained testing problems? Q: How to quantify the hardness of a constrained testing problem? Q: How does the hardness depend on the geometry? Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

15 Generalized Likelihood Ratio Test Generalized Likelihood Ratio Test (GLRT) { 1 if T (y) β φ β (y) : = 0 otherwise. where T (y) : = 2 log ( supθ C0 P θ (y) sup θ C1 P θ (y) ). Robertson and Wegman 78, Raubertas et al. 86, Fan et al Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

16 Generalized Likelihood Ratio Test Generalized Likelihood Ratio Test (GLRT) { 1 if T (y) β φ β (y) : = 0 otherwise. where Cone based GLRT: T (y) : = 2 log ( supθ C0 P θ (y) sup θ C1 P θ (y) T (y) = min θ C 0 y θ 2 2 min θ C 1 y θ 2 2 = Π C1 (y) 2 2 Π C0 (y) 2 2 ). Robertson and Wegman 78, Raubertas et al. 86, Fan et al Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

17 Related results GLRT different aspects of GLRT e.g. Wilks phenomenon, fitness of parametric model... [Warrack and Robertson 84, Menéndez 92, Lehmann 06, Perlman and Wu 99, Barlow 72, Lehamnn and Romano 06, Fan et al. 01, 07] Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

18 Related results GLRT different aspects of GLRT e.g. Wilks phenomenon, fitness of parametric model... [Warrack and Robertson 84, Menéndez 92, Lehmann 06, Perlman and Wu 99, Barlow 72, Lehamnn and Romano 06, Fan et al. 01, 07] Cone testing Raubertas, Lee & Nordheim studies the null distribution of GLRT Hu and Wright characterizes GLRT equivalences for various pairs of cones Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

19 Related results GLRT different aspects of GLRT e.g. Wilks phenomenon, fitness of parametric model... [Warrack and Robertson 84, Menéndez 92, Lehmann 06, Perlman and Wu 99, Barlow 72, Lehamnn and Romano 06, Fan et al. 01, 07] Cone testing Raubertas, Lee & Nordheim studies the null distribution of GLRT Hu and Wright characterizes GLRT equivalences for various pairs of cones Dümbgen 95, Robertson 78, Warrack and Robertson 84, Brown 86, Cohan and Sackrowitz 96, Rubertas et al 86, Menéndez 91 Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

20 Related results GLRT different aspects of GLRT e.g. Wilks phenomenon, fitness of parametric model... [Warrack and Robertson 84, Menéndez 92, Lehmann 06, Perlman and Wu 99, Barlow 72, Lehamnn and Romano 06, Fan et al. 01, 07] Cone testing Raubertas, Lee & Nordheim studies the null distribution of GLRT Hu and Wright characterizes GLRT equivalences for various pairs of cones Dümbgen 95, Robertson 78, Warrack and Robertson 84, Brown 86, Cohan and Sackrowitz 96, Rubertas et al 86, Menéndez 91 Minimax testing framework introduced in the seminal work of Ingster and co-authors different from the Neyman-Person testing framework Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

21 Testing radius Test θ C 0 v.s. C 1 \C 0 Uniform error for test ψ: error(ψ, ɛ) : = sup θ C 0 E θ [ψ(y)] }{{} type I error + sup θ C 1 \B 2 (ɛ;c 0 ) E θ [1 ψ(y)], } {{ } type II error Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

22 Testing radius Test θ C 0 v.s. C 1 \C 0 Uniform error for test ψ: error(ψ, ɛ) : = sup θ C 0 E θ [ψ(y)] }{{} type I error + sup θ C 1 \B 2 (ɛ;c 0 ) E θ [1 ψ(y)], } {{ } type II error Testing radius ɛ Ψ distance at which null/alternative are just distinguishable using class of tests Ψ Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

23 Minimax optimal testing radius Uniform error for test ψ: error(ψ, ɛ) : = sup θ C 0 E θ [ψ(y)] }{{} type I error + sup θ C 1 \B 2 (ɛ;c 0 ) E θ [1 ψ(y)], } {{ } type II error Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

24 Minimax optimal testing radius Uniform error for test ψ: error(ψ, ɛ) : = sup θ C 0 E θ [ψ(y)] }{{} type I error Minimax testing radius: ɛ OPT (ρ) : = inf + sup θ C 1 \B 2 (ɛ;c 0 ) E θ [1 ψ(y)], } {{ } type II error { ɛ inf ψ error(ψ, ɛ) ρ } [Ingster and Suslina 12, Ermakov 91, Lepski and Spokoiny 99, Lepski and Tsybakov 00] Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

25 Minimax optimal testing radius Uniform error for test ψ: error(ψ, ɛ) : = sup θ C 0 E θ [ψ(y)] }{{} type I error Minimax testing radius: ɛ OPT (ρ) : = inf + sup θ C 1 \B 2 (ɛ;c 0 ) E θ [1 ψ(y)], } {{ } type II error { ɛ inf ψ error(ψ, ɛ) ρ } [Ingster and Suslina 12, Ermakov 91, Lepski and Spokoiny 99, Lepski and Tsybakov 00] critical testing radius: ɛ GLRT and ɛ OPT Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

26 Main results Theorem (W, Wainwright & Guntuboyina 17) The GLRT testing radius satisfies ɛ 2 GLRT σ 2 min { width term, geometric term }. Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

27 Main results Theorem (W, Wainwright & Guntuboyina 17) The GLRT testing radius satisfies { ɛ 2 GLRT σ 2 min E Π C w 2, }{{} width term C 0 = 0, C 1 = C w N(0, I d ) Π C w : = arg min u C g u 2 S unit sphere in R d ( E Π C w ) 2 2 inf η, EΠ C w η C S }{{} geometric term }. Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22 supp

28 Main results Theorem (W, Wainwright & Guntuboyina 17) The GLRT testing radius satisfies { ɛ 2 GLRT σ 2 min E Π C w 2, }{{} width term ( E Π C w ) 2 2 inf η, EΠ C w η C S }{{} geometric term }. Width term = E Π C w 2 where Π C w : = arg min u C w u 2 E Π C (w) 2 = W(C S) where Gaussian width: W(A) : = E [ sup u, w ] u A Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

29 Main results Theorem (W, Wainwright & Guntuboyina 17) The GLRT testing radius satisfies Examples: { ɛ 2 GLRT σ 2 min E Π C w 2, }{{} width term ( E Π C w ) 2 2 inf η, EΠ C w η C S }{{} geometric term }. cone C width term geometric term k-dimensional space k non-negative orthant d d monotone cone log d ice cream cone d 1 Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

30 Ice Cream cone Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

31 Main results Theorem (W, Wainwright & Guntuboyina 17) The GLRT testing radius satisfies Examples: { ɛ 2 GLRT σ 2 min E Π C w 2, }{{} width term ( E Π C w ) 2 2 inf η, EΠ C w η C S }{{} geometric term }. cone C width term geometric term k-dimensional space k non-negative orthant d d monotone cone log d ice cream cone d 1 Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

32 Consequence for monotone cone H 0 : θ = θ 0, versus H 1 : θ M (monotone increasing sequence) θ 0 is k piece-wise constant optimal testing radius satisfies ( ) d ɛ 2 OPT(θ 0, M; ρ) σ k(θ 2 0 ) log k(θ 0 ) idea: H 0 : θ = 0, versus H 1 : θ T M (θ 0 ) tangent cone: T M (θ 0 ) : = {u R d t > 0 such that θ 0 + tu M }. Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

33 Minimax lower bound Theorem (W, Wainwright & Guntuboyina 17) The minimax testing radius satisfies ɛ 2 OPT σ 2 min { E Π C w 2, }{{} width term ( geometric term = GLRT is optimal whenever E Π C w 2 inf η C S η, EΠ C w ( ) 2 E Π C w 2 sup η, EΠ C w η C S ) 2 } {{ } geometric term II inf η, EΠ C w sup η, EΠ C w η C S η C S }. Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

34 Optimal hypothesis testing GLRT is minimax optimal (up to constants) in all these cases, cone C ɛ 2 GLRT k-dimensional space σ 2 k non-negative orthant σ 2 d monotone cone σ 2 log d ice cream cone σ 2 1 Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

35 Is GLRT always optimal? Consider a Cartesian product cone: Ice Cream d 1 (α) R Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

36 Is GLRT always optimal? Consider a Cartesian product cone: Ice Cream d 1 (α) R ɛ 2 GLRT σ 2 min{ d, } ɛ 2 OPT σ 2 min{ d, 1} Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

37 Is GLRT always optimal? Consider a Cartesian product cone: Ice Cream d 1 (α) R ɛ 2 GLRT σ 2 min{ d, } ɛ 2 OPT σ 2 min{ d, 1} a simple test based on (Y 1, Y d ) 2 minimax optimal supp Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

38 Summary show difficulty of testing depends on geometry (very different from estimation) Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

39 Summary show difficulty of testing depends on geometry (very different from estimation) interesting consequence for testing in monotone cone Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

40 Summary show difficulty of testing depends on geometry (very different from estimation) interesting consequence for testing in monotone cone the GLRT is NOT always optimal, and can be very poor even for simple problems Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

41 References Y. Wei, M. J. Wainwright, and A. Guntuboyina. (2017) The geometry of hypothesis testing over convex cones: Generalized likelihood tests and minimax radii.under revision by the Annals of Statistics. Thanks! Questions? Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22

42 Supplementary: general cones cone pairs (C 0, C 1 ) is said to be non-oblique if Π C0 (x) = Π C0 (Π C1 (x)) for all x R d. [Warrack et al. 84, Menéndez 92a,92b, Hu and Wright 94] C = C 0 C 1 polar cone of any C: C : = { v R d v, u 0 for all u C } Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22 back

43 Supplementary: sub-optimality of GLRT Consider a Cartesian product cone: Ice Cream d 1 (α) R Consider when signal lies only in the last coordinate: (0,..., 0, θ d ) Difference on GLRT under the null and the alternative: Π C (w 1, w 2,..., w d ) 2 Π C (w 1, w 2,..., θ d + w d ) 2 = (proj to ice cream, w d ) 2 (proj to ice cream, θ d + w d ) 2 where proj to ice cream = Π IC (w 1,..., w d 1 ) θ d proj to ice cream 2 back Yuting Wei (UC Berkeley) Geometric analysis of hypothesis testing Jan 30th, / 22