Bayesian Inference of Kinematics and Membership of Open Cluster

1. Bayesian Inference of Kinematics and Membership of Open Cluster Zhengyi Shao

2. Motivation: • Importance of Open Clusters: • Probe the structure and dynamics of MW • Average properties of a bunch of stars with similar Age, Met. and Distance • Dynamical of N-body systems with various StellarMass • Relatively easy to be identified: over density in Position, PM, Radial velocity and HR • Increased amount of Data: • Multiple bands of photometric survey, SDSS, 2MASS, UKIDSS … • Radial velocity survey, SEGUE, APOGEE, LAMOST … • Proper motion: GAIA • How to Build a framework based on Bayes method • Problems of combine different kinds of data, with different completeness • Focus on Kinematics and Memberships

3. Outlines • Introduction of Bayesian Inference Method • Bayesian Inference of Kinematics of Open Cluster • Method • Simulation

4. Bayesian Theory: • Probability:P(A), P(A | H) • Joint Probability: P(A,B | H) • Product rule: • The Bayesian Formula:

5. Bayesian formula and Data analysis: • Tasks of Data analysis: • Model selection • Parameter estimation • D : Observational data • M : Model • Θ : Parameters, Θ = (θ1,θ2,…,θn)

6. Posterior: • Likelihood: • Prior: π(Θ) • Probability Density Function of Θ , PDF(Θ) • Bayes Evidence:

7. The Likelihood: Compare model prediction: DM(Θ) vsDobs. Χ2

8. The prior: π(Θ) L(θ) π(θ) Post (θ) - π(θ) often assumed as uniform distribution of θorlnθ

9. Use prior to break the degeneracy between parameters • Example 1. H0, Ωk in cosmology • Example 2. t, met., dust in SED fitting of gals.

10. Parameter estimation:

11. Marginalized one-dimensional posterior probability distribution • - Project n-dim PDF, Post(Θ) to 1-dim of Post(θ1). • Set a new parameter f(Θ), project to Post(f(Θ)) • Statistical Values: Max, Med, Mean, SD ……

12. Bayesian Evidence: Bayes factor

13. Steps of Bayesian Inference • Collect the data, build the model & likelihood function • Consider the Prior of Θ • Cals. of L across all Prior  Posterior & Evidence • Problem of Cals: number of sampling ~ 10n • Metropolis–Hasting sampling • Nested sampling • Affine-invariant ensemble MCMC sampling

14. Nested Sampling & MultiNested Sampling

15. Summary of Bayesian Inference • Prior: • Marginalized posterior • Bayesian Evidence • Sampling Method

16. Kinematics of Open Clusters • HR (SED)  members  Kinematics • Solve the Mixture-Model of Cluster + Field Stars in the observational space of 2-dim position , proper motion and radial velocity. members  HR (SED)

17. Distribution of Cluster members & field stars

18. Mixture-Model of the distribution function Normalization integral: Likelihood function forthe i-th star, the likelihood is its probability of belonging this mixture model, so , for the whole sample: • Priors of parameters: • for central values, use uniform distribution • for scatters (σ), use uniform distribution in log • The number density of field stars, AF

19. Membership probability and its PDF

20. Simulation 500 cluster members + 1500 field stars

21. Test 1: Comparison of different combinations

22. Test 2. Off-central FoV

23. Test 3. incomplete obs. Joint likelihood：

24. Test 4: Comparison of different Likelihoods

25. Summary and Next step • Built the Likelihood of the Mixture-model with all position, PM, Rv, it is helpful to adding new kinds of Obs. Data. • The method is suitable for off-central FoV, and incomplete obs. • PDF of membership probabilities. • Test more complicated cases, elongation, segregation, substructures, et.al. • Apply to real clusters (M67 et.al.) • Apply to survey data to search and identify true clusters using Bayesian Evidence. • Other applications besides the OCs.