Embedding Supernova Cosmology into a Bayesian Hierarchical Model

Transcription

1 1 / 41 Embedding Supernova Cosmology into a Bayesian Hierarchical Model Xiyun Jiao Statistic Section Department of Mathematics Imperial College London Joint work with David van Dyk, Roberto Trotta & Hikmatali Shariff ICHASC Talk Jan 27, 2015

2 2 / 41 Outline 1 Algorithm Review 2 Combining Strategies 3 Surrogate Distribution 4 Extensions on Cosmological Model 5 Conclusion

3 1 Tanner, M. A. and Wong, W. H. (1987) 3 / 41 Problem Setting Goal: Sample from posterior distribution p(ψ Y ) using Gibbs-type samplers. Special case: Data Augmentation (DA) Algorithm 1 ψ = (θ, Y mis ). DA algorithm proceeds as: [Y mis θ ] [θ Y mis ]. Stationary distribution: p(y mis, θ Y ). DA algorithm and Gibbs samplers are easy to implement, but... Converge slowly!

4 1 Tanner, M. A. and Wong, W. H. (1987) 4 / 41 Problem Setting Goal: Sample from posterior distribution p(ψ Y ) using Gibbs-type samplers. Special case: Data Augmentation (DA) Algorithm 1 ψ = (θ, Y mis ). DA algorithm proceeds as: [Y mis θ ] [θ Y mis ]. Stationary distribution: p(y mis, θ Y ). DA algorithm and Gibbs samplers are easy to implement, but... Converge slowly!

5 Blue Rectangle Expand Paramter Space; Yellow Ellipse Change Conditioning Strategy 5 / 41 Algorithm Review MCMC Methods MDA (1999) DA Algorithm (1987) ASIS (2011) Gibbs Sampler (1993) PCG (2008)

6 6 / 41 Marginal Data Augmentation Marginal Data Augmentation (MDA) 2 MDA introduces a working parameter α into p(y, Y mis θ) via Y mis [e.g., Ỹmis = F α (Y mis )], s.t., p(ỹmis, Y θ, α)dỹmis = p(y θ). If the prior distribution of α is proper, MDA proceeds as: [α, Ỹmis θ ] [α, θ Ỹmis]. MDA improves convergence by increasing variability in augmented data and reducing augmented information. 2 Meng, X.-L. and van Dyk, D. A. (1999); Liu, J. S. and Wu, Y. N. (1999)

7 7 / 41 Ancillarity-Sufficiency Interweaving Strategy Ancillarity-Sufficiency Interweaving Strategy (ASIS) 3 ASIS considers a pair of special DA schemes: Sufficient augmentation Y mis,s : p(y Y mis,s, θ) is free of θ. Ancillary augmentation Y mis,a : p(y mis,a θ) is free of θ. Given θ, Y mis,a = F θ (Y mis,s ). ASIS proceeds as Interweave [θ Y mis,s ] into DA algorithm w.r.t. Y mis,a [Y mis,s θ ] [θ Y mis,s ] [Y mis,a Y mis,s, θ ] [θ Y mis,a ] [Y mis,s θ ] [Y mis,a Y mis,s ] [θ Y mis,a ] ASIS obtains more efficiency by taking advantage of the "beauty-and-beast" feature of two parent DA algorithms. 3 Yu, Y. and Meng, X.-L. (2011)

8 8 / 41 Understanding ASIS Model: Y (Y mis, θ) N(Y mis, 1), Y mis θ N(θ, V ), p(θ) 1. ASIS: Y mis,s = Y mis, Y mis,a = Y mis θ. [Y mis,s θ ] [θ Y mis,s ] [Y mis,a Y mis,s, θ ] [θ Y mis,a ] θ = constant Y mis = constant Y mis θ = constant Y mis Y mis Y mis θ θ θ More directions: efficient and easy to implement.

9 4 van Dyk, D. A. and Park, T. (2008) 9 / 41 Partially Collapsed Gibbs Sampling Partially Collapsed Gibbs (PCG) 4 Model Reduction: PCG reduces conditioning of Gibbs. It replaces some conditional distributions of a Gibbs sampler with conditionals of marginal distributions of the target. PCG improves convergence by increasing variance and jump size of conditional distributions. Three stages: Marginalization, permutation, trimming. Tools to transform a Gibbs sampler into a PCG one. Maintain the target stationary distribution.

10 10 / 41 Examples of PCG Sampling Example. ψ = (ψ 1, ψ 2, ψ 3, ψ 4 ); Sample from p(ψ Y ). Gibbs p(ψ 1 ψ 2, ψ 3, ψ 4 ) p(ψ 2 ψ 1, ψ 3, ψ 4 ) p(ψ 3 ψ 1, ψ 2, ψ 4 ) p(ψ 4 ψ 1, ψ 2, ψ 3 ) PCG I p(ψ 1 ψ 2, ψ 3, ψ 4 ) p(ψ 2, ψ 3 ψ 1, ψ 4 ) p(ψ 4 ψ 1, ψ 2, ψ 3 ) PCG II p(ψ 1 ψ 2, ψ 4 ) p(ψ 2, ψ 3 ψ 1, ψ 4 ) p(ψ 4 ψ 1, ψ 2, ψ 3 ) Special cases: blocked and collapsed Gibbs, e.g., PCG I. More interestingly, a PCG sampler consists of incompatible conditional distributions, e.g., PCG II. Modifying the order of steps of PCG II may alter its stationary distribution.

11 11 / 41 Three Stages to Derive a PCG Sampler (a) Gibbs p(ψ 1 ψ 2, ψ 3, ψ 4 ) p(ψ 2 ψ 1, ψ 3, ψ 4 ) p(ψ 3 ψ 1, ψ 2, ψ 4 ) p(ψ 4 ψ 1, ψ 2, ψ 3 ) (c) Permute p(ψ 1, ψ 3 ψ 2, ψ 4 ) p(ψ 2, ψ 3 ψ 1, ψ 4 ) p(ψ 2 ψ 1, ψ 3, ψ 4 ) p(ψ 4 ψ 1, ψ 2, ψ 3 ) (b) Marginalize p(ψ 1, ψ 3 ψ 2, ψ 4 ) p(ψ 2 ψ 1, ψ 3, ψ 4 ) p(ψ 2, ψ 3 ψ 1, ψ 4 ) p(ψ 4 ψ 1, ψ 2, ψ 3 ) (d) Trim [PCG II] p(ψ 1 ψ 2, ψ 4 ) p(ψ 2, ψ 3 ψ 1, ψ 4 ) p(ψ 4 ψ 1, ψ 2, ψ 3 ) Intermediate Draws

12 12 / 41 Outline 1 Algorithm Review 2 Combining Strategies 3 Surrogate Distribution 4 Extensions on Cosmological Model 5 Conclusion

13 5 Gilks et al. (1995) 6 van Dyk, D. A. and Jiao, X. (2015) 13 / 41 Combining Different Strategies into One Sampler Cannot Sample Conditionals? Embed Metropolis-Hastings (MH) into Gibbs 5 standard. Embed MH into PCG 6 subtle implementation! Further Improvement in Convergence Several parameters converge slowly a strategy is efficient for one parameter, but has little effect on others; Another strategy has opposite effect. By combining, we improve all. One strategy alone is useful for all parameters prefer to use a combination, as long as gained efficiency exceeds extra computational expense.

14 14 / 41 Background Physics Nobel Prize (2011): discovery of acceleration of expansion of the universe. The acceleration is attributed to existence of dark energy. Type Ia supernova (SNIa) observations: critical to quantify characteristics of dark energy. Mass > Chandrasekhar threshold (1.44 M ) = SN explosion. Image credit:

15 15 / 41 Standardizable Candles Common history = similar absolute magnitudes for SNIa, i.e., M i N(M 0, σint 2 ) = SNIa are standardizable candles. Phillips corrections: M i = Mi ɛ αx i + βc i, Mi ɛ N(M 0, σɛ 2 ); x i stretch correction, c i color correction, σ 2 ɛ σ 2 int

16 16 / 41 Distance Modulus Apparent Magnitude Absolute Magnitude = Distance Modulus: m B M = µ = 5log 10 [distance(mpc)] Nearby SN: distance = zc/h 0 ; Distant SN: µ = µ(z, Ω m, Ω Λ, H 0 ); c speed of light H 0 Hubble constant z redshift Ω m total matter density Ω Λ dark energy density

17 Bayesian Hierarchical Model 7 Level 1: Errors-in-variables regression: Level 2: ĉ i ˆx i ˆm Bi m Bi = µ i + M ɛ i αx i + βc i ; N c i x i m Bi, Ĉi, i = 1,..., n. M ɛ i N(M 0, σ 2 ɛ ); x i N(x 0, R 2 x ); c i N(c 0, R 2 c ). Priors: Gaussian for M 0, x 0, c 0 ; Uniform for Ω m, Ω Λ, α, β, log(r x ), log(r c ), log(σ ɛ ). z and H 0 fixed. 7 March et al. (2011) 17 / 41

18 18 / 41 Notation and Data Notation X (3n 1) (c 1, x 1, M ɛ 1,..., c n, x n, M ɛ n); b (3 1) (c 0, x 0, M 0 ); L (3n 1) (0, 0, µ 1,..., 0, 0, µ n ); T (3 3) = 0 1 0, and A (3n 3n) = Diag(T,..., T ). β α 1 Data: A sample of 288 SNIa compiled by Kessler et al. (2009).

19 19 / 41 Algorithms for Cosmological Herarchical Model MH within Gibbs sampler: Update of (Ω m, Ω Λ ) needs MH. MH within PCG sampler: Sample (Ω m, Ω Λ ) and (α, β) without conditioning on (X, b). Updates of both (Ω m, Ω Λ ) and (α, β) need MH. ASIS sampler: Y mis,s for (Ω m, Ω Λ ) and (α, β): AX + L; Y mis,a for (Ω m, Ω Λ ) and (α, β): X. MH within PCG+ASIS sampler: Given (α, β), sample (Ω m, Ω Λ ) with MH within PCG; Given (Ω m, Ω Λ ), sample (α, β) with ASIS. For each sampler, run 11,000 iterations with a burn-in of 1,000.

20 Convergence Results of Gibbs and PCG Ωm MH within Gibb MH within PCG Ωm ΩΛ 1.2 ΩΛ 1.2 α β α β / 41

21 Convergence Results of ASIS and Combining Ωm ASIS PCG within ASIS Ωm ΩΛ 1.2 ΩΛ 1.2 α β α β / 41

22 22 / 41 Effective Sample Size (ESS) per Second The larger the ESS/sec, the more efficient the algorithm. Gibbs PCG ASIS PCG+ASIS Ω m Ω Λ α β

23 23 / 41 Factor Analysis Model Model Y i N [ ] Z i β, Σ = Diag(σ1 2,..., σ2 p), for i = 1,..., n. Y i (1 p) vector of the ith observation; Z i (1 q) vector of factors; Z i β N(0, I); q < p. β and Σ unknown parameters. Priors: p(β) 1; σ 2 j Inv-Gamma(0.01, 0.01), j = 1,..., p. Simulation Study Set p = 6, q = 2, and n = 100. σ 2 j Inv-Gamma(1, 0.5), (j = 1,..., 6); β hj N(0, 3 2 ), (h = 1, 2; j = 1,..., 6).

24 Algorithms for Factor Analysis Standard Gibbs sampler: [ Z β, Σ ] [ ] σj 2 Z, β, σ j 2 p [β Z, Σ]. j=1 MH within PCG sampler: sampling σ 2 1, σ2 3 and σ2 4 without conditioning on Z. This should be facilitated by MH. ASIS sampler: Y mis,a for β: Z i ; Y mis,s for β: W i = Z i β. MH within PCG+ASIS sampler: Given β, update Σ with MH within PCG; Given Σ, update β with ASIS. For each sampler, run 11,000 iterations with a burn-in of 1, / 41

25 Convergence Results of Factor Analysis Model Gibbs 2 log(σ 1 ) PCG 2 log(σ 1 ) ASIS 2 log(σ 1 ) Gibbs β PCG β ASIS β PCG+ASIS 2 log(σ 1 ) PCG+ASIS β / 41

26 26 / 41 Effective Sample Size (ESS) per Second The larger the ESS/sec, the more efficient the algorithm. Gibbs PCG ASIS PCG + ASIS log(σ1 2 ) β

27 27 / 41 Outline 1 Algorithm Review 2 Combining Strategies 3 Surrogate Distribution 4 Extensions on Cosmological Model 5 Conclusion

28 28 / 41 Bivariate Surrogate Distribution Target distribution: p(ψ 1, ψ 2 ). Surrogate distribution: π(ψ 1, ψ 2 ). π(ψ 1 ) = p(ψ 1 ), π(ψ 2 ) = p(ψ 2 ); The correlation between ψ 1 and ψ 2 is lower for π than for p. Sampler S.1 p(ψ 1 ψ 2 ) p(ψ 2 ψ 1 ) Sampler S.2 π(ψ 1 ψ 2 ) p(ψ 2 ψ 1 ) Sampler S.3 π(ψ 1 ψ 2 ) π(ψ 2 ψ 1 ) Stationary distribution of Samplers S.1 and S.2: p(ψ 1, ψ 2 ). Stationary distribution of Sampler S.3: π(ψ 1, ψ 2 ). Condition for Sampler S.2 maintaining the target: π(ψ 1 ) = p(ψ 1 ), π(ψ 2 ) = p(ψ 2 ); Step order is fixed.

29 29 / 41 Comparison of Samplers S.1 S.3 Example. [( 0 p(ψ 1, ψ 2 ) : N 0 ) ( , )] [( 0 ; π(ψ 1, ψ 2 ) : N 0 ) ( 1 ρπ, ρ π 1 )]. Convergence rate Sampler S.1 Sampler S.2 Sampler S.3 ρ π

30 30 / 41 Ways to Derive Surrogate Distributions ASIS: [Y mis,s θ ] [Y mis,a Y mis,s ] [θ Y mis,a ]. π(θ Y mis,s ) = p(y mis,a Y mis,s )p(θ Y mis,a )dy mis,a ; π(θ, Y mis,s ) = π(θ Y mis,s )p(y mis,s ). PCG: intermediate stationary distributions. PCG II: [ψ 1 ψ 2, ψ 4 ] [ψ 2, ψ 3 ψ 1, ψ 4 ] [ψ 4 ψ 1, ψ 2, ψ 3 ]. Intermediate stationary ending with Step 1: π(ψ 1, ψ 2, ψ 3, ψ 4 ) = p(ψ 2, ψ 3, ψ 4 )p(ψ 1 ψ 2, ψ 4 ). MDA: [α, Ỹmis θ ] [α, θ Ỹmis]. p(θ Ỹmis) = p(α, θ Ỹmis)dα Set Ỹmis as Y ========== mis π(θ Y mis ); π(θ, Y mis ) = π(θ Y mis )p(y mis ).

31 31 / 41 Advantages of Surrogate Distribution Surrogate distribution unifies different strategies under a common framework. For ASIS, a sampler involving surrogate distribution, but equivalent to the original ASIS sampler, has fewer steps. If we are only interested in marginal distributions, surrogate distribution strategy is promising to produce more efficient algorithms.

32 32 / 41 Outline 1 Algorithm Review 2 Combining Strategies 3 Surrogate Distribution 4 Extensions on Cosmological Model 5 Conclusion

33 Model Review and New Data Recall: Level 1: Errors-in-variables regression: Level 2: ĉ i ˆx i ˆm Bi m Bi = µ i + M ɛ i αx i + βc i ; N c i x i m Bi, Ĉi, i = 1,..., n. M ɛ i N(M 0, σ 2 ɛ ); x i N(x 0, R 2 x ); c i N(c 0, R 2 c ). σ ɛ small = Standardizable candle Data: A JLA sample of 740 SNIa in Betoule, et al. (2014). 33 / 41

34 34 / 41 Shrinkage Estimation Low mean squared error estimates of M ɛ i Conditional Posterior Expectation of M i ε % CI σ ε

35 35 / 41 Shrinkage Error Reduced standard deviations Conditional Posterior Standard Deviation of M i ε % CI σ ε

36 36 / 41 Systematic Errors Systematic errors: seven sources of uncertainties. Blocks: different surveys. Effect on cosmological parameters: Ĉ stat vs Ĉstat + Ĉsys Ω Λ % stat+sys 95% stat+sys 68% only stat 95% only stat Ω m

37 Adjusting for Galaxy Mass: Method I Method I: Divide M ɛ i by w i = log 10 (M galaxy /M ); { M ɛ i N(M 01, σ 2 ɛ1 ), if w i < 10, M ɛ i N(M 02, σ 2 ɛ2 ), if w i > wi Density posterior mean of M i ε Density 37 / 41

38 Adjusting for Galaxy Mass: Method II Much scatter in both M i and w i w i posterior mean of M i ε Treat w i as covariate like x i and c i, ŵ i N(w i, ˆσ 2 w): m Bi = µ i + M ɛ i αx i + βc i + γw i γ Density 38 / 41

39 Model Checking Model setting: Fix (Ω m,ω Λ ); m Bi = µ i + M ɛ i αx i + βc i ; µ i = µ(z i, Ω m, Ω Λ, H 0 )+t(z i ), t(z i ) cubic spline Results: Red line posterior mean; Gray band 95% region; Black dots ( ˆm Bi M 0 + αˆx i βĉ i ) µ i z (red shift) µ mean µtheory Cubic Spline Curve Fitting (K=4) 39 / 41

40 40 / 41 Outline 1 Algorithm Review 2 Combining Strategies 3 Surrogate Distribution 4 Extensions on Cosmological Model 5 Conclusion

41 41 / 41 Conclusion Summary Combining strategy and surrogate distribution samplers are useful to produce more efficiency in convergence. The hierarchical Gaussian model reflects the underlying physical understanding of supernova cosmology. Future Work More numerical examples to illustrate the algorithms. Complete the theory of surrogate distribution strategy. Embed this hierarchical model into a model for the full time-series of the supernova explosion, using Gaussian process to impute apparent magnitudes over time.