Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/Stat310_0809/pb_20080930.pdf Дата изменения: Tue Sep 30 23:24:49 2008 Дата индексирования: Tue Oct 2 04:16:30 2012 Кодировка: Поисковые слова: m 5

Motivation

Mo delling the CMD

Making Inference

Results

Conclusion

Ages of stellar populations from color-magnitude diagrams Paul Baines
Department of Statistics Harvard University

September 30, 2008

Paul Baines

093008


Motivation Context & Example

Mo delling the CMD

Making Inference

Results

Conclusion

Welcome! Today we will look at using hierarchical Bayesian modeling to make inference about the properties of stars; most notably the age and mass of groups of stars. Complete with a brief dummies (statisticians) guide to the Astronomy behind it.

Paul Baines

093008



Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Isochrones For Dummies/Statisticians

Given the mass, age and metallicity of a star, we `know' what its `ideal' observation should be i.e., where it should be on the CMD. The tables of these `ideal' observations are called isochrone tables. Why are they only `ideal' colours/magnitudes?

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Observational Error

Alas, as with every experiment there are observational errors and biases caused by the instruments.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Observational Error

Alas, as with every experiment there are observational errors and biases caused by the instruments. 1. These are relatively well understood ­ and can be considered to be Gaussian with known standard deviation.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Observational Error

Alas, as with every experiment there are observational errors and biases caused by the instruments. 1. These are relatively well understood ­ and can be considered to be Gaussian with known standard deviation. 2. Importantly, we can characterize the standard deviation as a function of the observed data. i.e., given Yi = (YiB , YiV , YiI )T we have i = (Yi ).

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

The Observed Data
We observe (depending on the experiment) p different colours/magnitudes for n stars. Although it is equally straightforward to model colours U - B , B - V etc., and magnitudes B , V , etc., we will stick with magnitudes. The (known) standard deviations in each band are also recorded for each observation. We also observe that we observe the n stars in the dataset and that we didn't observe any others!

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

The Likelihood I
yi = Where, ~ fi =
1
(B ) i

Bi

i = 1, . . . , n (1)

1

(V ) i

1 (I i

)

~ Vi Ai , Mi , Z N fi , R Ii

1 Bi 1 Vi 1 Ii

· fb (Ai , Mi , Z ) · fv (Ai , Mi , Z ) , · fi (Ai , Mi , Z )



1

(

BV )

R = (BV ) (BI )

1
(VI )

(BI ) (VI ) 1

.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

The Likelihood II
Let Si = 1 if star i is observed, Si = 0 otherwise. Si |Yi Bernoulli (p (Yi )) (2)

where p (Yi ) is the probability of a star of a given magnitude being unobserved (provided by Astronomers).

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

The Likelihood II
Let Si = 1 if star i is observed, Si = 0 otherwise. Si |Yi Bernoulli (p (Yi )) (2)

where p (Yi ) is the probability of a star of a given magnitude being unobserved (provided by Astronomers). Note: We can also have Si = (SiB , SiV , SiI )T and allow for some stars to be observed only in a subset of the bands.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

The Parameters

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Mass

Before we have any data, the prior distributions of mass and age are still not independent. We know a priori that old stars cannot have large mass, likewise for very young stars. Hence, we specify the prior on mass conditional on age: p (Mi |Ai , M i.e. Mi |Ai , M
min

, Mmax (Ai ) , )
max

1 ·1 Mi {Mi

[Mmin ,Mmax (Ai )]}

(3)

min

,M

(Ai ) , Truncated-Pareto.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Age

For age we assume the following hierarchical structure:
2 2 Ai |µA , A N µA , A iid

(4)

where Ai = log

10

2 (Age), with µA and A hyperparameters. . .

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Metallicity

Denoted by Zi . Assumed to be known and common to all stars i.e., Zi = Z = 4

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Hyperparameters

Next, we model the hyperparameters with the simple conjugate form:
2 µA |A N µ0 , 2 A 0

,

2 A Inv -

2

2 0 , 0

(5)

2 Where µ0 , 0 , 0 and 0 are fixed by the user to represent prior knowledge (or lack of ).

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Correlation

We assume a uniform prior over the space of positive definite correlation matrices. This isn't quite uniform on each of ( very close.
BV )

, (BI ) and (

VI )

, but it is

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Incompleteness

Unfortunately, some dimmer stars may not be fully observed. This censoring can bias conclusions about the stellar cluster parameters.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Incompleteness

Unfortunately, some dimmer stars may not be fully observed. This censoring can bias conclusions about the stellar cluster parameters. Since magnitudes are functions of photon arrivals, the censoring is stochastic.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Putting it all together
Sij |Yi Bernoulli (p (Yi )) i = 1, . . . , n, n + 1, . . . , n + nmis j {B , V , I } 1 (B ) Bi i 1 ~ i = 1, . . . , n, n + 1, . . . , n + nmis yi = (V ) Vi Ai , Mi , Z N fi , R i 1 (I ) Ii
i

Mi |Ai , M

min

, Truncated-Pareto ( - 1, M Ai |µA ,
2 iid A

min

,M

max

(Ai ))

N µA ,

2 A 2 2 0

2 µA | A N

µ0 ,

2 A 0

,

2 A Inv - {Rp .d .}

0 ,

p (R) 1

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Observed-Data Posterior
The product of the densities on the previous slide gives us the complete-data posterior. Alas, we don't observe all the stars, and nmis is an unknown parameter. For now, lets just condition on nmis . We have: W
obs

= =

n, y

[1:n]

= (y1 , . . . , yn ) , S = {1, . . . , 1, 0, . . . , 0)} ,M
[(n+1):(n+m)]

Wmis

m, Y

[(n+1):(n+m)]

,A

[(n+1):(n+m)]

= where X
a :b

M[1:n] , A

2 [1:n] , µA , A

,R
+1

denotes the vector (Xa , Xa

, . . . , Xb )

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Observed-Data Posterior

We want p (|Wobs ) but so far we have p (, W integrate out the missing data: p (|Wobs ) =

mis

|W

obs

). So, we

p (, Wmis |Wobs ) d Wmis

(6)

In practice, this integration is done by sampling from p (, Wmis |Wobs ) and retaining only the samples of .

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Observed-Data Posterior

We form a Gibbs sampler to sample from p (, Wmis |Wobs ). Given (t ) a current state of our Markov Chain, = (t ) and Wmis = Wmis : 1. Draw (t
+1)

from p |Wmis , Wobs from p Wmis |(t
+1)

(t )

(as before)
obs

2. Draw Wmis

(t +1)

,W

(new)

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling Wmis
At each iteration of the Gibbs sampler we need to draw the missing data from the appropriate distribution. In other words, given a bunch of masses, ages, and metallicities of nmis missing stars, find a bunch of Yi 's that are consistent with that:
pi Yi |Y[1:n] , M, A, µA , exp - 1 ~ Yi - f (Yi ; Mi , Ai , Z ) 2
T 2 A

[1 - (Yi )] ·
-1

(7) (8)

R

~ Yi - f (Yi ; Mi , Ai , Z )

for i = n + 1, . . . , n + m.

Paul Baines

093008


Motivation Intro duction & Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling Wmis

Once we have sampled a new set of Ymis , we need to sample the standard deviation of the Gaussian error for those stars. Here we assume this is a deterministic mapping: = (Yi ).

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling from the posterior
Some notes:

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling from the posterior
Some notes: 1. We have our model ­ what does our posterior look like?

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling from the posterior
Some notes: 1. We have our model ­ what does our posterior look like? 2. Ugly. No chance of working with it analytically MCMC!

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling from the posterior
Some notes: 1. We have our model ­ what does our posterior look like? 2. Ugly. No chance of working with it analytically MCMC! 3. Going to have to be a Gibbs sampler. How to break it up?

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling from the posterior
Some notes: 1. We have our model ­ what does our posterior look like? 2. Ugly. No chance of working with it analytically MCMC! 3. Going to have to be a Gibbs sampler. How to break it up? 4. Mass and Age are going to be extremely highly correlated (i.e., sample jointly)

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling from the posterior
Some notes: 1. We have our model ­ what does our posterior look like? 2. Ugly. No chance of working with it analytically MCMC! 3. Going to have to be a Gibbs sampler. How to break it up? 4. Mass and Age are going to be extremely highly correlated (i.e., sample jointly) 5. No analytic simplification for terms in Mi , Ai because of f

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Sampling from the posterior
Some notes: 1. We have our model ­ what does our posterior look like? 2. Ugly. No chance of working with it analytically MCMC! 3. Going to have to be a Gibbs sampler. How to break it up? 4. Mass and Age are going to be extremely highly correlated (i.e., sample jointly) 5. No analytic simplification for terms in Mi , Ai because of f 6. High dimensional multi-modal, so we also use parallel tempering.

Paul Baines

093008


Motivation The Algorithm

Mo delling the CMD

Making Inference

Results

Conclusion

Parallel Tempering

A brief overview of parallel tempering: The parallel tempering framework involves sampling N chains, with the i th chain of the form: pi () = p (|y)1/ti exp - H () ti (9)

As ti increases the target distributions become flatter.

Paul Baines

093008


Motivation Do es it work?

Mo delling the CMD

Making Inference

Results

Conclusion

Simulation Results

We simulate 100 datasets from the model with n = 100:
2 µA = 9.2 A = 0.012

M(min) = 0.8 = 2.5 R=I (Bi , Vi , Ii ) (0.03, 0.12)

Paul Baines

093008


mu_age: Posterior medians vs. Truth

9.5

9.4

q qq q q qq q q q q q qq q q qq q qq qq q q qq q q qq qq qq q q qq q q q q q q q qq q q q q q q q qq q q qq q q q qq qq q q qqq q qq q q q q q qq q q q qq q q q

q

True(mu_age)

9.3

q

9.1

9.2

9.0

q q q

8.9 8.9

9.0

9.1

9.2

9.3

9.4

9.5

Median(mu_age)


Motivation Do es it work?

Mo delling the CMD

Making Inference

Results

Conclusion

Post_p m_cover a_cover mu_age ss_age

0.5 0.6 0.4 3.0 0.0

1.0 1.2 1.1 3.0 0.0

2.5 2.8 2.8 5.0 3.0

5.0 6.0 6.3 6.0 4.0

25.0 25.0 25.1 30.0 26.0

50.0 49.1 50.4 55.0 47.0

Paul Baines

093008


Motivation Do es it work?

Mo delling the CMD

Making Inference

Results

Conclusion

Post_p m_cover a_cover mu_age ss_age

50.0 49.1 50.4 55.0 47.0

75.0 74.1 75.3 81.0 71.0

95.0 94.4 93.7 97.0 94.0

97.5 99.0 99.5 96.6 98.5 99.2 97.2 99.0 99.3 99.0 100.0 100.0 97.0 100.0 100.0

Paul Baines

093008


Nominal vs. Actual Coverage
100

Target m_i a_i mu_age ss_age

Nominal 0 0 20 40

60

80

20

40 Actual

60

80

100


Motivation

Mo delling the CMD

Making Inference

Results

Conclusion

Future Work & Conclusions

Future Work

Some important things still need to be built into the model before it is fit for purpose: Extinction/Absorption: Shift in observed data Multi-Cluster Mo dels: Allow for multiple stellar clusters

Paul Baines

093008