Using what we know: Inference with Physical Constraints

Transcription

1 Using what we know: Inference with Physical Constraints Chad M. Schafer & Philip B. Stark Department of Statistics University of California, Berkeley PhyStat September 2003 Some work joint w/ Steve Evans (Berkeley) Ben Hansen (Michigan)

2 Examples of Physical Constraints Nonnegativity: Counts, energies, densities; monotonicity Bounds on functionals: Energy in geomagnetic field, rotation rate in solar interior (w/ C. Genovese) Parametrizations: Power law for CMB spectrum (w/ L. Tenorio, C. Lineweaver) How can we use such constraints to reduce uncertainty? Most texts don t treat constraints. 1

3 Examples. Transparencies. Seismic velocity in Earth s core (w/ Bob Parker) aftershock probability density (w/ N. Hengartner) SH coefficients of CMB and of geomagnetic field 2

4 Models and Parameters Model specifies prob. distribution of data X. Models indexed by θ; call θ and Pr θ the model. Physical Constraint: Know a priori that θ Θ Parameter: image f(θ) of θ under a mapping f. Building block: Bounded Normal Mean Model X N(θ, 1), with θ Θ = [ τ, τ] Seek to estimate f(θ) = θ. 3

5 Confidence Sets and Coverage Confidence Set: Random set S of parameter values: depends on X Coverage probability (1 α): Minimum chance S(X) contains true f(θ). Pr θ {S(X) f(θ)} 1 α, θ Θ. (1) Duality between testing and confidence sets: Invert family of level α tests 1 α conf. set all θ Θ that aren t rejected. Standard 95% conf. interval for normal mean: I = [X 1.96, X ]. Doesn t use the constraint θ [ τ, τ]. Procrustean (truncated) interval: I T = [X 1.96, X ] [ τ, τ]. Uses constraint, but is it best? 4

6 Common ways to add constraints Ignore them. (Prof. Barlow claimed that s the only frequentist option) Ad hoc; procrustean: Make unconstrained estimate; force it to be in the constraint set. Bayesian: Use prior π that assigns probability 1 to the constraint set. Pr{θ Θ} = 1. (2) π Frequentist minimax: Use estimator that (within some class) is minimax for some loss over the constraint set 5

7 Shortcomings Ad hoc; procrustean: Can do better. Bayesian: Where does the prior come from? No such thing as uninformative prior. Sensitive to loss and assumptions Frequentist minimax: Driven by worst case Sensitive to loss and assumptions 6

8 Defining best: small expected size Want a precise answer: Minimize expected size of the confidence set. Expected size depends on true value of θ: tradeoff. Bayesian: minimize average (for prior π) expected size Frequentist Minimax: minimize max expected size for θ Θ Bayes/Minimax duality: Minimax is Bayes for least favorable prior 7

9 Minimax expected size CI for bdd Normal mean standard truncated Best meas. Opt. standard fixed-width a meas. b τ I I [ τ, τ] I N I OPT % % % % % % % % % % % 3.3 6% % % 3.3 3% % % 3.3 1% % % 3.3 0% % % 3.5 5% 3.3 c % % 3.6 6% % % 3.6 5% 3.4 a These have form [ˆθ(X) e, ˆθ(X) + e], with ˆθ( ) measurable and e constant. b These have form {θ Θ : (θ, X) S}, where S Θ X is product-measurable. c Truncated Pratt interval I TP is optimal for τ The entries in the rightmost column for τ = 3.50, 3.75, and 4.00 are approximated numerically. 8

10 Size isn t everything: how you use it matters. Cost of including θ θ can depend on θ: E.g., might sacrifice length to include values with one sign only. Curvature of the universe? Can allow size measure to depend on θ. Numerics: Approximate least-favorable π by Monte Carlo Well suited to distributed/parallel computing. Uses importance sampling. Don t need closed-form likelihood. Solve 2-player matrix game iteratively. 9

11 Example: Microwave cosmology relic of the Big Bang Model for Power Spectrum X N 0, N + l=1 ( 2l + 1 4π ) C l (θ)b 2 l P l P l : the l th Legendre polynomial matrix N: noise covariance matrix {C l (θ)}: power spectrum for the set of cosmological parameters θ B l : transfer function of the observing filter. Complicated relationship between cosmological parameters and spectrum {C l }: nonlinear PDE. 10

12 Constraints on Parameters of Cosmological Model Parameter Lower Upper Total Matter Ω m Baryonic Matter Ω b Cosmological Constant Ω Λ Hubble Constant (km s 1 Mpc 1 ) H Scalar Spectral Index n s Optical Depth τ Relative to critical density. Monte-Carlo choses model uniformly, subject to Ω b Ω m and 0.6 Ω m + Ω Λ 1.4. (3) 11

13 MAXIMA-1 Data polar angle (degrees) µk azimuthal angle (degrees) 12

14 Data reduction/compression Form 20,000 linear combinations of 5972 pixel temps. Select 5972 linear combinations eigenvectors of covariance matrix for reference model. Select 2000 of those using estimated Kullbach-Leibler divergence to approximate false coverage prob. 13

15 Results 100 WMAP (Bennett, et.al. 2003) MAXIMA 1 (Hanany, et.al. 2000) 90 `(` + 1)C` /2π (µk) q Angular Frequency (`) 35 of 1000 models not rejected. Spectra of 300 rejected models shown. 14

16 Results l(l + 1)C l /2π (µk) Angular Frequency (l)

17 Results (contd) Location of main peak (l): (175, 235). Height of main peak (µk): (59.7, 75.5). Location of first valley (l): (348, 468). Height of first valley (µk): (22.3, 35.6). 16

18 Results (contd) Some Accepted Models: Ω b Ω m Ω Λ τ H 0 n s Ω Min incl Max incl Min poss Max poss

19 References Balbi, A., et al., Constraints on Cosmological Parameters from MAXIMA-1, Astrophys. J., 545, L1 L4 Bennett, C.L., et al., First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Maps and Basic Results, Ap. J., in press. Evans, S.N., B. Hansen, and P.B. Stark, Minimax Expected Measure Confidence Sets for Restricted Location Parameters. Tech. Rept. 617, Dept. Statistics Univ. Calif Berkeley. Hanany, et al., MAXIMA-1: A Measurement of the Cosmic Microwave Background Anisotropy on angular scales of 10 arcminutes to 5 degrees, Astrophys. J., 545, L5. 18

20 Jaffe, et al., Cosmology from Maxima-1, Boomerang and COBE/DMR CMB Observations, Phys. Rev. Lett., 86, Kempthorne, P.J., Numerical specification of discrete least favorable prior distributions, SIAM J. sci. statis. Comp., 8, Nelson, W., Minimax solution of statistical decision problems by iteration, Ann. math. Statis., 37, Pratt, J.W., Length of Confidence Intervals, J. amer. statis. Assoc., 56, Rao, C.R., Efficient estimates and optimum inference procedures, J. roy. statis. Soc., 24 (B), Robinson, J., An iterative method for solving a game, Ann. Math., 54,