Luminosity Functions

Transcription

1 Department of Statistics, Harvard University May 15th, 2012

2 Introduction: What is a Luminosity Function? Figure: A galaxy cluster.

3 Introduction: Project Goal The Luminosity Function specifies the relative number of sources at each luminosity.

4 Introduction: Project Goal The Luminosity Function specifies the relative number of sources at each luminosity. Goal of the Project: To develop a fully Bayesian model to infer the distribution of the luminosities of all the sources in a population.

5 Introduction: Data Y i, observed photon counts, contaminated with background in a source exposure. X, observed photon counts in the exposure of pure background.

6 Level I model: X ξ Pois(ξ), Y i = Y ib + Y is, where Y ib ξ Pois(a i ξ), { δ 0, if λ i = 0; Y is λ i Pois(b i λ i ) Pois(b i λ i ), if λ i 0. ξ is the background intensity, λ i is the intensity of source i, a i is ratio of source area to background area (known constant), b i is the telescope effective area (known constant).

7 Level II model: ξ Gamma(α 0, β 0 ), { = 0, with probability 1 π; λ i α, β, π Gamma(α, β), with probability π. Level III model: P(α, β, π) 1 β 3 πc 1 1 (1 π) c 2 1.

8 : Summary Model X ξ Pois(ξ), Y i = Y ib + Y is, Y is ξ Pois(a i ξ), Y ib λ i Pois(b i λ i ), ξ Gamma(α 0, β 0 ), { = 0, with probability 1 π, λ i α, β, π Gamma(α, β), with probability π, P(α, β, π) 1 β 3. Research interest: The posterior distribution of intensities λ, The posterior distribution of 1 π, the proportion of dark sources.

9 Simulation Study 1: Setup Number of sources, N=500. Distribution of λ s: λ i iid { δ 0, with prob 1 π = 0.15, Gamma[10, 35] with prob Distribution of Y is : Y is λ i Pois(b i λ i ). Distribution of background noise Y ib : Y i = Y is + Y ib Observed Photon Counts Y ib ξ Pois(4), approximately.

10 Posterior Distributions of the Hyper-parameters posterior draws of alpha/beta posterior draws of alpha/beta^2 posteriro draws of 1-pi Density Density Density

11 Z Posterior Distributions of the Hyper-parameters Scatter Plots of Posterior Draws Joint Posterior Density of pi and alpha/beta The True Value Posterior Draws pi pi alpha/beta alpha/beta

12 Distribution of source intensities Distribution of source intensities λ i iid { δ 0, with prob 1 π, Gamma(α, β), with prob π

13 Simulation Study 2: Setup Number of sources, N=500. Distribution of λ s: λ i iid { δ 0, with prob 1 π = 0.15, Gamma[2, 60] with prob Distribution of Y is : Y is λ i Pois(b i λ i ). Y ib ξ Pois(2), approximately Observed Photon Counts

14 Simulation Study 2 posterior draws of alpha/beta posterior draws of alpha/beta^2 posteriro draws of 1-pi Density Density Density

15 Z Z Z Z Simulation Study 2 Scatter Plots of Posterior Draws pi The True Value Posterior Draws alpha/beta pi alpha/beta pi pi alpha/beta pi alpha/beta alpha/beta

16 Simulation Study 2 Distribution of source intensities

17 Choices of Priors for the Hyper-parameters Recall λ i α, β, π iid { δ0, with prob 1 π, Gamma[ α β, α ] = Gamma[µ, µ β 2 β ] with prob π.

18 Choices of Priors for the Hyper-parameters Recall λ i α, β, π iid { δ0, with prob 1 π, Gamma[ α β, α ] = Gamma[µ, µ β 2 β ] with prob π. P(µ, β, π)dµdβdπ P(β)P(π)dµdβdπ 1 β P(β)P(π)dαdβdπ, P(α, β, π)dαdβdπ 1 β c 3+1 πc 1 1 (1 π) c 2 1 dαdβdπ

19 Choices of Priors: Simulation Study π Beta(c 1, c 2 ), we can use informative prior if we have some prior information about the distribution of π. Otherwise, we can let c 1 = c 2 = 1, so π Unif (0, 1). Priors for (α, β): Prior 0: P(α, β) 1, Prior 1: P(α, β) 1 β, Prior 2: P(α, β) 1 β 2, Prior 3: P(α, β) 1 β 3.

20 Choices of Priors: Coverage for π prior0 prior1 prior2 prior3 Gamma[2,60] Gamma[4,100] Gamma[8,200] Gamma[2,20] Gamma[4,30] Gamma[8,50] Gamma[2,2] Gamma[4,5] Gamma[8,8]

21 Choices of Priors: Coverage for α β prior0 prior1 prior2 prior3 Gamma[2,60] Gamma[4,100] Gamma[8,200] Gamma[2,20] Gamma[4,30] Gamma[8,50] Gamma[2,2] Gamma[4,5] Gamma[8,8]

22 Choices of Priors Conclusion: the prior P(α, β) 1 β 3 gives the highest frequency coverage in most simulation studies. This prior is called Stein s Harmonic Prior. The SHP prior is shown to provide estimators that have adequate frequency coverage.

23 Speeding up MCMC It takes about 3 hours to get 100,000 draws. Metropolis-Hastings algorithm within Gibbs Sampler: P(α β, π, ξ, λ, Y B, X, Ỹ ) where K = 1 λi 0, and λ = ( i,λ i 0 λ i) 1/K. ( (βλ ) α ) K, Γ(α) Is there a good way to sample from the conditional posterior distribution?

24 Acknowledgement Vinay Kashyap, David van Dyk, Andreas Zezas.