Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/Stat310_1415/jxy_20150127.pdf Дата изменения: Tue Jan 27 16:15:11 2015 Дата индексирования: Sun Apr 10 12:05:16 2016 Кодировка: Поисковые слова: m 35

Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Embedding Supernova Cosmology into a Bayesian Hierarchical Model
Xiyun Jiao
Statistic Section Depar tment of Mathematics Imperial College London

Joint work with David van Dyk, Rober to Trotta & Hikmatali Shariff

ICHASC Talk Jan 27, 2015

1 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Outline

1

Algorithm Review Combining Strategies Surrogate Distribution Extensions on Cosmological Model Conclusion

2

3

4

5

2 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Problem Setting
Goal: Sample from posterior distribution p( |Y ) using Gibbs-type samplers. Special case: Data Augmentation (DA) Algorithm1 = (, Ymis ). DA algorithm proceeds as: [Ymis | ] - [|Ymis ]. Stationar y distribution: p(Ymis , |Y ).

DA algorithm and Gibbs samplers are easy to implement, but. . .

Converge slowly!
1

Tanner, M. A. and Wong, W. H. (1987)
3 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Problem Setting
Goal: Sample from posterior distribution p( |Y ) using Gibbs-type samplers. Special case: Data Augmentation (DA) Algorithm1 = (, Ymis ). DA algorithm proceeds as: [Ymis | ] - [|Ymis ]. Stationar y distribution: p(Ymis , |Y ).

DA algorithm and Gibbs samplers are easy to implement, but. . .

Converge slowly!
1

Tanner, M. A. and Wong, W. H. (1987)
4 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Algorithm Review
MCMC Methods

MDA (1999)

DA Algorithm (1987)

ASIS (2011)

Gibbs Sampler (1993)

PCG (2008) Blue Rectangle--Expand Paramter Space; Yellow Ellipse--Change Conditioning Strategy
5 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Marginal Data Augmentation
Marginal Data Augmentation (MDA)2 MDA introduces a working parameter into p(Y , Ymis |) ~ via Ymis [e.g., Ymis = F (Ymis )], s.t., ~ p(Y
mis

~ , Y |, )dYmis = p(Y |).

If the prior distribution of is proper, MDA proceeds as: ~ ~ [ , Ymis | ] - [, |Ymis ]. 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)
6 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Ancillarity-Sufficiency Interweaving Strategy
Ancillarity-Sufficiency Interweaving Strategy (ASIS)3 ASIS considers a pair of special DA schemes:
Sufficient augmentation Ymis,S : p(Y |Ymis,S , ) is free of . Ancillar y augmentation Ymis,A : p(Ymis,A |) is free of .

Given , Y

mis,A

= F (Ymis,S ). ASIS proceeds as

Interweave [ |Ymis,S ] into DA algorithm w.r.t. Ymis,A

[Ymis,S | ] [ |Ymis,S ] [Ymis,A |Ymis,S , ] [|Ymis,A ] [Ymis,S | ] [Ymis,A |Ymis,S ] [|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)
7 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Understanding ASIS
Model: Y |(Ymis , ) N(Ymis , 1), Ymis | N(, V ), p() 1. ASIS: Ymis,S = Ymis , Ymis,A = Ymis - . [Ymis,S | ] [ |Y
= constant
3 3

mis,S

] [Ymis,A |Ymis,S , ] [|Ymis,A ]
Ymis = constant
3

Ymis - = constant

2

2

s

s

Ymi 1

Ymi 1

0

0

-1

-1

-2

-1

0

1

2

3

4

-2

-1

0

1

2

3

4

-1
-2

0

Ymi 1

s

2

-1

0

1

2

3

4

More directions: efficient and easy to implement.
8 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Par tially 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 stationar y distribution.

4

van Dyk, D. A. and Park, T. (2008)
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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Examples of PCG Sampling
Example. = (1 , 2 , 3 , 4 ); Sample from p( |Y ). Gibbs |2 , 3 |1 , 3 |1 , 2 |1 , 2 PCG I p(1 |2 , 3 , 4 ) p(2 , 3 |1 , 4 ) p(4 |1 , 2 , 3 ) PCG II p(1 |2 , 4 ) p(2 , 3 |1 , 4 ) p(4 |1 , 2 , 3 )

p p p p

( ( ( (

1 2 3 4

, , , ,

4 4 4 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.

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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Three Stages to Derive a PCG Sampler
(a) Gibbs (1 |2 , 3 , 4 (2 |1 , 3 , 4 (3 |1 , 2 , 4 (4 |1 , 2 , 3 Permute , 3 |2 , 4 , 3 |1 , 4 |1 , 3 , 4 |1 , 2 , 3 (b) Marginalize p(1 , 3 |2 , 4 ) p(2 |1 , 3 , 4 ) p(2 , 3 |1 , 4 ) p(4 |1 , 2 , 3 ) (d) Trim [PCG II] p(1 |2 , 4 ) p(2 , 3 |1 , 4 ) p(4 |1 , 2 , 3 )

p p p p

) ) ) )

p p p p

(c) (1 (2 (2 (4

) ) ) )

" "--Intermediate Draws
11 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Outline

1

Algorithm Review Combining Strategies Surrogate Distribution Extensions on Cosmological Model Conclusion

2

3

4

5

12 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Combining Different Strategies into One Sampler
Cannot Sample Conditionals? Embed Metropolis-Hastings (MH) into Gibbs5 --standard. Embed MH into PCG6 --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.
5 6

Gilks et al. (1995) van Dyk, D. A. and Jiao, X. (2015)
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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

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: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/snovcn.html

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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

"Standardizable Candles"
Common histor y = similar absolute magnitudes for SNIa, i.e.,
2 Mi N(M0 , int )

= SNIa are "standardizable candles". Phillips corrections: Mi = Mi - xi + ci , Mi N(M0 , 2 ); xi --stretch correction, ci --color correction, 2
2 int

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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Distance Modulus
Apparent Magnitude - Absolute Magnitude = Distance Modulus: mB - M = µ = 5log10 [distance(Mpc)] + 25. Nearby SN: distance = zc /H0 ; Distant SN: µ = µ(z , m , , H0 );
c --speed of light H0 --Hubble constant z --redshift m --total matter density --dark energy density

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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Bayesian Hierarchical Model7
Level 1: Errors-in-variables regression: = µi + Mi - xi + ci ; ci ^ N xi , Ci , i = 1, . . . , n. mBi m
Bi



^ ci xi ^ ^ Bi m Level 2:

2 2 Mi N(M0 , 2 ); xi N(x0 , Rx ); ci N(c0 , Rc ).

Priors: Gaussian for M0 , x0 , c0 ; Uniform for m , , , , log(Rx ), log(Rc ), log( ). z and H0 fixed.
7

March et al. (2011)
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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Notation and Data

Notation X(3 b(3 L(3
nв1)

--(c1 , x1 , M1 , . . . , cn , xn , Mn ); µ1 , . . . , 0, 0, µn ); 00 1 0 , and A( - 1

в1)

--(c0 , x0 , M0 ); --(0, 0, 1 = 0

nв1)

T(3

в3)

3nв3n)

= Diag(T , . . . , T ).

Data: A sample of 288 SNIa compiled by Kessler et al. (2009).

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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Algorithms for Cosmological Herarchical Model
MH within Gibbs sampler: Update of (m , ) needs MH. MH within PCG sampler:
Sample (m , ) and (, ) without conditioning on (X , b). Updates of both (m , ) and (, ) need MH.

ASIS sampler: Ymis,S for (m , ) and (, ): AX + L; Ymis,A for (m , ) and (, ): X . MH within PCG+ASIS sampler:
Given (, ), sample (m , ) with MH within PCG; Given (m , ), sample (, ) with ASIS.

For each sampler, run 11,000 iterations with a burn-in of 1,000.
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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Convergence Results of Gibbs and PCG
MH within Gibb
0.6 Autocorrelation 0.0 0.4 0.8 0.6 m 0.2 0.4 m 0.2 0.4

MH within PCG
Autocorrelation 0.0 0.4 0.8

0.0

0
1.2

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40
1.2

0.0

0

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40

Autocorrelation 0.0 0.4 0.8

0.0

0
0.18

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40
0.18

0.0

0

2000

4000 6000 Iteration

8000 10000

Autocorrelation 0.0 0.4 0.8

0.4 0.8

0.4 0.8

0

10

20 Lag

30

40

Autocorrelation 0.0 0.4 0.8

0.06

0

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40

0.06

0

2000

4000 6000 Iteration

8000 10000

Autocorrelation 0.0 0.4 0.8

0.12

0.12

0

10

20 Lag

30

40

Autocorrelation 0.0 0.4 0.8

0

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40

0

2000

4000 6000 Iteration

8000 10000

Autocorrelation 0.0 0.4 0.8

3.0

2.6

2.2

2.2

2.6

3.0

0

10

20 Lag

30

40

20 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Convergence Results of ASIS and Combining
ASIS
0.6 Autocorrelation 0.0 0.4 0.8 0.6 m 0.2 0.4 m 0.2 0.4

PCG within ASIS
Autocorrelation 0.0 0.4 0.8

0.0

0
1.2

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40
1.2

0.0

0

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40

Autocorrelation 0.0 0.4 0.8

0.0

0
0.18

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40
0.18

0.0

0

2000

4000 6000 Iteration

8000 10000

Autocorrelation 0.0 0.4 0.8

0.4 0.8

0.4 0.8

0

10

20 Lag

30

40

Autocorrelation 0.0 0.4 0.8

0.06

0

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40

0.06

0

2000

4000 6000 Iteration

8000 10000

Autocorrelation 0.0 0.4 0.8

0.12

0.12

0

10

20 Lag

30

40

Autocorrelation 0.0 0.4 0.8

0

2000

4000 6000 Iteration

8000 10000

0

10

20 Lag

30

40

0

2000

4000 6000 Iteration

8000 10000

Autocorrelation 0.0 0.4 0.8

3.0

2.6

2.2

2.2

2.6

3.0

0

10

20 Lag

30

40

21 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Effective Sample Size (ESS) per Second
The larger the ESS/sec, the more efficient the algorithm. Gibbs m


PCG 0.0302 0.0232 0.0556 0.0264

ASIS 0.0103 0.00571 0.0787 0.0830

PCG+ASIS 0.0392 0.0282 0.0826 0.0733

0.00166 0.000997 0.00712 0.00874



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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Factor Analysis Model
Model
2 2 Yi N Zi , = Diag(1 , . . . , p ) , for i = 1, . . . , n.

Yi --(1 в p) vector of the i th obser vation; Zi --(1 в q ) vector of factors; Zi | N(0, I ); q < p. and --unknown parameters. Priors: p( ) 1; j2 Inv-Gamma(0.01, 0.01), j = 1, . . . , p.

Simulation Study
Set p = 6, q = 2, and n = 100. j2 Inv-Gamma(1, 0.5), (j = 1, . . . , 6); hj N(0, 32 ), (h = 1, 2; j = 1, . . . , 6).

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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Algorithms for Factor Analysis
Standard Gibbs sampler:
2 Z | , - j2 |Z , , -j p j =1

- [ |Z , ].

2 2 2 MH within PCG sampler: sampling 1 , 3 and 4 without conditioning on Z . This should be facilitated by MH.

ASIS sampler: Ymis,A for : Zi ; Ymis,S for : Wi = Zi . MH within PCG+ASIS sampler:
Given , update with MH within PCG; Given , update with ASIS.

For each sampler, run 11,000 iterations with a burn-in of 1,000.
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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Convergence Results of Factor Analysis Model
-1 Autocorrelation 0.4 0.8 2 Autocorrelation 0.4 0.8

Gibbs

Gibbs

log(2 1 -5 -3

)

0.0

-2

-1

13 0

1

-7

0

2000

4000 6000 Iteration

8000

10000

0

10

20 Lag

30

40

0

2000

4000 6000 Iteration

8000

10000

0.0

0

10

20 Lag

30

40

-1

Autocorrelation 0.4 0.8

2

0.0

-2

-7

0

2000

4000 6000 Iteration

8000

10000

0

10

20 Lag

30

40

0

2000

4000 6000 Iteration

8000

10000

0.0

Autocorrelation 0.4 0.8

PCG

PCG

log(2 1 -5 -3

)

-1

13 0

1

0

10

20 Lag

30

40

-1

Autocorrelation 0.4 0.8

2

0.0

-2

-7

0

2000

4000 6000 Iteration

8000

10000

0

10

20 Lag

30

40

0

2000

4000 6000 Iteration

8000

10000

0.0

Autocorrelation 0.4 0.8

ASIS

ASIS

log(2 1 -5 -3

)

-1

13 0

1

0

10

20 Lag

30

40

PCG+ASIS

PCG+ASIS

-1

Autocorrelation 0.4 0.8

2

0.0

-2

-7

0

2000

4000 6000 Iteration

8000

10000

0

10

20 Lag

30

40

0

2000

4000 6000 Iteration

8000

10000

0.0

Autocorrelation 0.4 0.8

log(2 1 -5 -3

)

-1

13 0

1

0

10

20 Lag

30

40

25 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Effective Sample Size (ESS) per Second
The larger the ESS/sec, the more efficient the algorithm.

Gibbs

PCG

ASIS

PCG + ASIS

2 log(1 )

0.18

2.17

0.15

1.91



13

0.0087

0.0090

17.54

15.37

26 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Outline

1

Algorithm Review Combining Strategies Surrogate Distribution Extensions on Cosmological Model Conclusion

2

3

4

5

27 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Bivariate Surrogate Distribution
Target distribution: p(1 , 2 ). Surrogate distribution: (1 , 2 ). (1 ) = p(1 ), (2 ) = p(2 ); The correlation between 1 and 2 is lower for than for p. Sampler S.1 p(1 |2 ) p(2 |1 ) Sampler S.2 (1 |2 ) p(2 |1 ) Sampler S.3 (1 |2 ) (2 |1 )

Stationar y distribution of Samplers S.1 and S.2: p(1 , 2 ). Stationar y distribution of Sampler S.3: (1 , 2 ). Condition for Sampler S.2 maintaining the target: (1 ) = p(1 ), (2 ) = p(2 ); Step order is fixed.
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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Comparison of Samplers S.1­S.3
Example.
p(1 , 2 ) : N
0 0

,

1 0.99

0.99 1

; (1 , 2 ) : N

0 0

,

1

1

.

Sampler S.1 Sampler S.2 Sampler S.3

Convergence rate

0.0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.2

0.4

0.6

0.8

29 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Ways to Derive Surrogate Distributions
ASIS: [Ymis,S | ] [Ymis,A |Ymis,S ] [|Ymis,A ]. (|Ymis,S ) = p(Ymis,A |Ymis,S )p(|Ymis,A )dYmis,A ; (, Ymis,S ) = (|Ymis,S )p(Ymis,S ). PCG: intermediate stationary distributions. PCG II: [1 |2 , 4 ] [2 , 3 |1 , 4 ] [4 |1 , 2 , 3 ]. Intermediate stationary ending with Step 1: (1 , 2 , 3 , 4 ) = p(2 , 3 , 4 )p(1 |2 , 4 ). ~ ~ MDA: [ , Ymis | ] - [, |Y
mis

].

~ ~ p(|Ymis ) = p(, |Ymis )d = = = = = = (|Ymis ); = = = = == (, Ymis ) = (|Ymis )p(Ymis ).
30 / 41

~ Set Ymis as Ymis


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

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.

31 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Outline

1

Algorithm Review Combining Strategies Surrogate Distribution Extensions on Cosmological Model Conclusion

2

3

4

5

32 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Model Review and New Data
Recall: Level 1: Errors-in-variables regression: = µi + Mi - xi + ci ; ci ^ N xi , Ci , i = 1, . . . , n. mBi m
Bi



^ ci xi ^ ^ mBi Level 2:

2 2 Mi N(M0 , 2 ); xi N(x0 , Rx ); ci N(c0 , Rc ).

small = "Standardizable candle" Data: A "JLA" sample of 740 SNIa in Betoule, et al. (2014).
33 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Shrinkage Estimation
Low mean squared error estimates of Mi
-18.6

95% CI

Conditional Posterior Expectation of M i

-19.6

-19.4

-19.2

-19.0

-18.8

0.00

0.05

0.10

0.15

0.20

34 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Shrinkage Error
Reduced standard deviations
0.20

95% CI

Conditional Posterior Standard Deviation of M

i

0.00

0.05

0.10

0.15

0.00

0.05

0.10


0.15

0.20

35 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Systematic Errors
Systematic errors: seven sources of uncer tainties. Blocks: different sur veys.
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.0 0.2 0.4




Effect on cosmological parameters: ^ ^ ^ Cstat vs Cstat + Csys .

68% 95% 68% 95%
0.6
m

stat+sys stat+sys only stat only stat
1.0

0.8

36 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Adjusting for Galaxy Mass: Method I
Method I: Divide Mi by wi = log10 (Mgalaxy /M ); Mi N(M01 , 21 ), if wi < 10, Mi N(M02 , 22 ), if wi > 10.
-18.8 -18.8 -18.8
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q qq q q q q q qqq q qq q q q q q qq q q q q qq q qq q q q q q qq q q q q q qqq q q q q q q q q q qq qq q q qq q q q q q q q qqqq q qq q qq q q q q qq q q qq q q qq qq q q q q q q q q q qq q q q q q qq q qq q q q qq qq q q q qq q qq q q q qqq qq qqq q q q q q qq q qq qq q qq q q qq q q qq qq qq qqqqq qqqqqq q qq q qq q q q q q q q q q q q q q q q q qq q q q q qq q q qq q q q qq q q q q q q qq q qqq qq q q q q qqqq qqq q q q qq qq q qq qq q q q q q qq q qqqq q qq qq qq q q qqq q qqq q q q q q qq q q q q q q q q q q q qqqqq qq qqq qq qq q qqq q qq q q q qq q q q q q q q qqq q q q q qq qqqqq qqqq q qqqqq qq q q q q q q q qq qqq q q q q q q qq q qqqqq qq qqq q qq qq qq q q qqq qq q q q q q q qqq q qq q q qq q q q qq qq qqqqqqqq qq qq q q q qq qq q qq q q q q q qq q qq q qq qqqqq qqq q q q qq qq q q q q qq q q q q qq q q q q q qqq qq q q q q q q q q q q q q q q q q q q qq q qq q q q q qq q q q q qqqq q qq q qq q qqq q q qqq q q qqq q q q qq qq q q q qq q qq q q q q q qq qq q qq qq qq q q q qq q q q q q qq q q qq q q q

-18.9

-18.9

-19.0

-19.0

posterior mean of Mi

-19.1

-19.1

-19.2

-19.2

-19.3

-19.3

-19.4

-19.4

0

2

4

6

8

10 12

5

6

7

8 w
i

9

10

11

12

-19.4

-19.3

-19.2

-19.1

-19.0

-18.9

0

2

4

6

8

10

12

Density

Density

37 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Adjusting for Galaxy Mass: Method II
Much scatter in both Mi and wi . Treat wi as covariate like xi and ci , ^ wi N(wi , w ): ^2 mBi = µi + Mi - xi + ci + wi .

-18.9

-18.8

q q q q q qq q q q q q qq q qq qq q q q q q q qq qq qq q q qq qqq q qq q q qqq q qq qq qq q q q qqq q q q qqq qqq q q q q q q qq qqq qq q q qqqq qq q qq q q q qq q q q q q qq q qqq qqqqq q qqq q qq qqqq q q q qqq q qq q q q qq qq qqq q q qqq q q q qqqqq q qq q q q q q qq qq q q qqq q q q qq q qq q q qq q q q qq q q qqqq qqqqqqqqq qqq qq q qqqq qqq qq qqq q qq qqqq q q q qqqqq qqqqqqqq q q qq q q qq q qq q q qq qqqqqqqqq qq q q q q qqq qqq qqqq qqq q q q q q q qqqqqqqq q q qq q qqqqqqqq q q qqqqqqqqq q qq qqq qq qqqq qqqqqqq qq qqqqq qqqq q q q qqqqq q q qqq qqqq q q qq q q qqq qqqqqq qqqq qq q q q qq q q qq qq q q qq q q qq qqq qq qqq qqqq q qq qqqqqq q q q qqqq q qqqqqqqqqq qqqqq q q qq q qqqq qqq qq qq q q qq qq q q q qqqqqqq q q qq q qq q q qq q qq q q q qq q qq q qq qq qq q q q qqq qq q q q q

posterior mean of M

-19.0

q q q q q q q q q qq q q q q

-19.1

-19.2

q q q q q q q q q q q

Density

-19.4

-19.3

4

6

8 wi

10

12

0

10

20

30

i

-0.05

0.00

0.05

0.10

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Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Model Checking
Model setting: Fix (m , ); mBi = µi + Mi - xi + ci ; ~ µi = µ(zi , m , , H0 )+t (zi ), ~
theory

Cubic Spline Curve Fitting (K=4)

0.4
q q q q q qq q q q qq qqqq q q q qq q q q qq q q qq qq q q q q qq q q q q qq q q qq q qq q q qq qq q qq q qq q q qq q q q q qq qq q qq q qq q q q q qqq q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q qq q q q q q qq q q q q qqq q q q qqq q q q q q qq q q q qqq q q q q q qqq q q q q q q q qqq q qqq q q qq q qq q q q q qq q qq q qqq q q q qq q q qq q q q qq q q q q q qq q q q q q qqqqqq q q q q qq q q qq q q q q q qq q q q qqqqq q qq q q q q q q q qqq qq q q q qq qq qq q qq qq qqqqq qqqqq q q q q qq q q q q q qq q q q q q q q q qqq q q qq q q qq q q qq qq q qq q q q q q qq q q q q q q q qq q q qq q q qq qqqqq qqq qqq q qq q qq q q q qq q q q q qq qq q q q q q qqq q q qq qq q qq qq qqq q qq q qq q q q qqqqq qq q q q q q q q qq qq qq qq q q q qq q q q qq q q q qqq q qq q q qqqq qq q q q qq qq q q q qq qq q qqq q qq q q q q q q q q q qq q qqq q q qq q qq q q qq qq qq q qqq q q qq qq q q q q q q q qq qqq qqqqq qq qq q q q q qq q q q q qq q q q q q q qq q q q q q q q qq qq q q q q q q qq qq q q q q q q q qq q q qq qqq q qq q q q q q q qqq q q q qq q q q q q qq q q qq qq q q q q q q q

-µ

0.2

t (zi )--cubic spline Results: Red line--posterior mean; Gray band--95% region; Black dots-- ^ ^ ^ (mBi - M0 + xi - ci ) - µi . ~

q

µm

ean

-0.2

0.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z (red shift)

39 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

Outline

1

Algorithm Review Combining Strategies Surrogate Distribution Extensions on Cosmological Model Conclusion

2

3

4

5

40 / 41


Algorithm Review

Combining Strategies

Surrogate Distribution

Extensions on Cosmological Model

Conclusion

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 theor y 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.

41 / 41