Ages of stellar populations from color-magnitude diagrams. Paul Baines. September 30, 2008

Transcription

1 Ages of stellar populations from color-magnitude diagrams Paul Baines Department of Statistics Harvard University September 30, 2008

2 Context & Example 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.

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4 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?

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

6 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.

7 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 Y i = (Y ib, Y iv, Y ii ) T we have σ i = σ (Y i ).

8 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!

9 The Likelihood I y i = 1 B σ (B) i i 1 V σ (V ) i i 1 I σ (I ) i i ) A i, M i, Z N ( f i, R i = 1,..., n (1) Where, f i = 1 σ Bi f b (A i, M i, Z) 1 σ Vi f v (A i, M i, Z) 1 σ Ii f i (A i, M i, Z), R = 1 ρ (BV ) ρ (BI ) ρ (BV ) 1 ρ (VI ) ρ (BI ) ρ (VI ) 1.

10 The Likelihood II Let S i = 1 if star i is observed, S i = 0 otherwise. S i Y i Bernoulli (p (Y i )) (2) where p(y i ) is the probability of a star of a given magnitude being unobserved (provided by Astronomers).

11 The Likelihood II Let S i = 1 if star i is observed, S i = 0 otherwise. S i Y i Bernoulli (p (Y i )) (2) where p(y i ) is the probability of a star of a given magnitude being unobserved (provided by Astronomers). Note: We can also have S i = (S ib, S iv, S ii ) T and allow for some stars to be observed only in a subset of the bands.

12 The Parameters

13 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 (M i A i, M min, M max (A i ), α) 1 M α i 1 {Mi [M min,m max (A i )]} (3) i.e. M i A i, M min, M max (A i ), α Truncated-Pareto.

14 Age For age we assume the following hierarchical structure: A i µ A, σa 2 iid ( N µa, σa) 2 (4) where A i = log 10 (Age), with µ A and σ 2 A hyperparameters...

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

16 Hyperparameters Next, we model the hyperparameters with the simple conjugate form: ) µ A σa (µ 2 N 0, σ2 A, σa 2 κ Inv ( χ2 ν 0, σ 2 ) 0 (5) 0 Where µ 0, κ 0, ν 0 and σ0 2 knowledge (or lack of). are fixed by the user to represent prior

17 Correlation We assume a uniform prior over the space of positive definite correlation matrices. This isn t quite uniform on each of ρ (BV ), ρ (BI ) and ρ (VI ), but it is very close.

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

19 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.

20 Putting it all together S ij Y i Bernoulli (p (Y i )) i = 1,..., n, n + 1,..., n + n mis j {B, V, I } y i = 1 B σ (B) i i 1 V σ (V ) i i 1 I σ (I ) i i ) A i, M i, Z N ( f i, R i = 1,..., n, n + 1,..., n + n mis M i A i, M min, α Truncated-Pareto (α 1, M min, M max (A i )) ( ) A i µ A, σa 2 iid N µ A, σa 2 ( ) ( ) µ A σa 2 N µ 0, σ2 A, σa 2 Inv χ 2 ν 0, σ0 2 κ 0 p(r) 1 {Rp.d.}

21 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 n mis is an unknown parameter. For now, lets just condition on n mis. We have: W obs = { n, y [1:n] = (y 1,..., y n ), S = {1,..., 1, 0,..., 0)} } W mis = { m, Y [(n+1):(n+m)], M [(n+1):(n+m)], A [(n+1):(n+m)] } Θ = { M [1:n], A [1:n], µ A, σ 2 A, R} where X a:b denotes the vector (X a, X a+1,..., X b )

22 Observed-Data Posterior We want p (Θ W obs ) but so far we have p (Θ, W mis W obs ). So, we integrate out the missing data: p (Θ W obs ) = p (Θ, W mis W obs ) dw mis (6) In practice, this integration is done by sampling from p (Θ, W mis W obs ) and retaining only the samples of Θ.

23 Observed-Data Posterior We form a Gibbs sampler to sample from p (Θ, W mis W obs ). Given a current state of our Markov Chain, Θ = Θ (t) and W mis = W (t) mis ( ) : 1. Draw Θ (t+1) from p Θ W (t) mis, W obs (as before) 2. Draw W (t+1) mis from p ( W mis Θ (t+1), W obs ) (new)

24 Sampling W mis 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 n mis missing stars, find a bunch of Y i s that are consistent with that: ) p i (Y i Y [1:n], M, A, µ A, σa 2 [1 π (Y i )] (7) { exp 1 ( Y i 2 f T (Y i ; M i, A i, Z)) ( R 1 Y i f (Y i ; M i, A i, Z)) } (8) for i = n + 1,..., n + m.

25 Sampling W mis Once we have sampled a new set of Y mis, we need to sample the standard deviation of the Gaussian error for those stars. Here we assume this is a deterministic mapping: σ = σ (Y i ).

26 The Algorithm Sampling from the posterior Some notes:

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

28 The Algorithm 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!

29 The Algorithm 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?

30 The Algorithm 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)

31 The Algorithm 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 M i, A i because of f

32 The Algorithm 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 M i, A i because of f 6. High dimensional multi-modal, so we also use parallel tempering.

33 The Algorithm Parallel Tempering A brief overview of parallel tempering: The parallel tempering framework involves sampling N chains, with the i th chain of the form: { p i (θ) = p(θ y) 1/t i exp H(θ) } (9) t i As t i increases the target distributions become flatter.

34 Does it work? Simulation Results We simulate 100 datasets from the model with n = 100: µ A = 9.2 σa 2 = M (min) = 0.8 α = 2.5 R = I (σ Bi, σ Vi, σ Ii ) (0.03, 0.12)

35 mu_age: Posterior medians vs. Truth Median(mu_age) True(mu_age)

36 Does it work? Post_p m_cover a_cover mu_age ss_age

37 Does it work? Post_p m_cover a_cover mu_age ss_age

38 Nominal vs. Actual Coverage Nominal Target m_i a_i mu_age ss_age Actual

39 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 Models: Allow for multiple stellar clusters