Refining Bayesian Data Analysis Methods for Use with Longer Waveforms

1. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Reﬁning Bayesian Data Analysis Methods for use with Longer Waveforms An investigation of parallelization of the "nested sampling" algorithm and the application of variable resolution functions James Michael Bell Millsaps College University of Florida IREU in Gravitational-Wave Physics James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

2. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Compact Binaries Motivation Research Objectives Coalescing Compact Binaries Black Hole and Neutron Star Pairs Primary candidate for ground-based GW detectors. Expected rate of occurrence (per Mpc3Myr) NS-NS: 0.01 to 10 NS-BH: 4 × 10−4 to 1 BH-BH: 1 × 10−4 to 0.3 Implications of ﬁndings Further validation of general relativity Insight about physical extrema James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

3. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Compact Binaries Motivation Research Objectives Motivation Technology & Limits Initial LIGO and Virgo detectors Signal visibility ~30s Advanced Detector Configuration Signal Visibility >3min Increased Efficiency ⇒ Increased Data Use ⇒ More Significant Results James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms

4. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Compact Binaries Motivation Research Objectives Motivation Recent Progress in the Field "Nested Sampling" (2004) J. Skilling Bayesian coherent analysis of in-spiral gravitational wave signals with a detector network (2010) J. Veitch and A. Vecchio James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

5. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Compact Binaries Motivation Research Objectives Research Objectives Primary To investigate increased parallelization of the existing nested sampling algorithm Secondary To develop a variable resolution algorithm that will improve the handling of template waveforms James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

6. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Compact Binaries Motivation Research Objectives Research Objectives Primary To investigate increased parallelization of the existing nested sampling algorithm Secondary To develop a variable resolution algorithm that will improve the handling of template waveforms James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

7. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Bayes’ Theorem Parameter Estimation Model Selection Bayes’ Theorem Derivation H = {Hi|i = 1, ..., N} ⊂ I and D ⊂ H James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

9. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Bayes’ Theorem Parameter Estimation Model Selection Parameter Estimation Identifying the parameters Hypothesis depends on a minimum of 9 parameters Θ = {M, ν, t0, φ0, DL, α, δ, ψ, ι} 2 masses, time, sky position, distance, 3 orientation angles Other possible parameters 2 magnitudes and 4 orientation angles for spins 2 parameters for the equation of state More? James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

10. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Bayes’ Theorem Parameter Estimation Model Selection Parameter Estimation Marginalization Goals: Find the distribution of each parameter Find the expectation of each parameter Two Parameter Marginalization James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

11. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Bayes’ Theorem Parameter Estimation Model Selection Parameter Estimation Marginalization Procedure Let −→ θA ⊂ Θ so that θ ≡ {θA, θB}, θA,B ∈ ΘA,B. Calculate the marginalized distribution p( −→ θ A| −→ d , H, I) = ΘB p( −→ θ A| −→ d , H, I)d −→ θ B Determine the mean expected value −→ θ A = ΘA −→ θ Ap( −→ θ A| −→ d , H, I)d −→ θ A James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

14. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Bayes’ Theorem Parameter Estimation Model Selection Model Selection Bayesian Hypothesis Testing The Bayes Factor P(Hi|I)P( −→ d |Hi, I) P(Hj|I)P( −→ d |Hj, I) = P(Hi|I) P(Hj|I) K K H Support Strength < 1 j ? 1-3 i Weak 3-10 i Substantial 10-30 i Strong 30-100 i Very Strong > 100 i Decisive James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

15. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Bayes’ Theorem Parameter Estimation Model Selection Model Selection Quantifying the Evidence Calculating the Evidence Integral Z = P( −→ d |Hi, I) = −→ θ ∈Θ p( −→ d | −→ θ , Hi, I)p( −→ θ |Hi, I)d −→ θ Computational Problems Dimensionality of Θ Large intervals to integrate James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

16. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Bayes’ Theorem Parameter Estimation Model Selection Model Selection Quantifying the Evidence Calculating the Evidence Integral Z = P( −→ d |Hi, I) = −→ θ ∈Θ p( −→ d | −→ θ , Hi, I)p( −→ θ |Hi, I)d −→ θ Computational Problems Dimensionality of Θ Large intervals to integrate James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

17. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The objective of nested sampling What we need: To calculate the evidence integral using random sample What we want: To reduce time of evidence computations To produce marginalized PDFs and expectations To increase accuracy of previous algorithms James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

18. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling Bayes’ Theorem Revisited P(Hi| −→ d , I) = P( −→ d |Hi, I)P(Hi|I) P( −→ d |I) P( −→ d |θ, I) P(θ|I) = P( −→ d |I) P(θ| −→ d , I) Likelihood × Prior = Evidence × Posterior L(θ)× π(θ) = Z× P(θ) James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

19. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Procedure 1 Map Θ to R1. James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

20. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Procedure 2 Draw N samples {Xi|i = 1...N} from π(x) and ﬁnd L(x). James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

21. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Procedure 3 Order {xi|i = 1...N} from greatest to least L. James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

22. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Procedure 4 Remove Xj corresponding to Lmin. James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

23. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Procedure 5 Store the smallest sample Xj and its corresponding L(x). James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

24. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Procedure 6 Draw Xi+1 ∈ U(0, Xi) to replace Xi corresponding to Lmin. James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

25. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Procedure 7 Repeat, shrinking {Xi} to regions of increasing likelihood. James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

26. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Result 8 Area Z = 1 0 L(x)δx ≈ 1 0 L(x)dx shown in (a). James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

27. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Result 9 Sample from Area Z → Sample from P(x) = L(x)/Z James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

28. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling The Result 10 Sample from P(x) = L(x)/Z ⇒ Sample from P( −→ x | −→ d , I) James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

29. Introduction Data Analysis Nested Sampling Variable Resolution Appendices The Algorithm The Result Parallelization Nested Sampling Parallelization of the Existing Algorithm Run algorithm in parallel with different random seeds Save each sample set and its likelihood values Collate the results of the multiple runs Sort the resulting samples by their likelihood values Treat samples as part of a collection {NT } = Nruns k=1 Nk Each parallel run contains Nk live points Re-apply nested sampling with lower sample weight James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

36. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Motivation The Goal Brainstorming Variable Resolution Motivation Improved efﬁciency with better template waveform handling Higher resolution ⇒ Increased computation time Lower resolution ⇒ Decreased accuracy Current algorithm utilizes stationary resolution function James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

37. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Motivation The Goal Brainstorming Variable Resolution The Goal Implement a variable resolution function Exploit the monochromatic nature of the early waveform Focus computational resources on more complex regions James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

38. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Motivation The Goal Brainstorming Variable Resolution Brainstorming Possible Methods Time-series variation of least-squares parameters Event triggering Monte Carlo Methods and/or further nested sampling James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

39. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

40. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix A: Sources 1 arXiv:0911.3820v2 [astro-ph.CO] 2 Data Analysis: A Bayesian Tutorial; D.S. Sivia with J. Skilling 3 http://www.stat.duke.edu/~fab2/nested_sampling_talk.pdf 4 http://www.mrao.cam.ac.uk/ steve/malta2009/images/ nestposter.pdf 5 http://ba.stat.cmu.edu/journal/2006/vol01/issue04/ skilling.pdf 6 Dr. John Veitch and Dr. Chris Van Den Broeck James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

41. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix A: Sources 7 http://www.inference.phy.cam.ac.uk/bayesys/ 8 http://arxiv.org/pdf/0704.3704.pdf 9 Dr. Shadow J.Q. Robinson, Millsaps College 10 Dr. Mark Lynch, Millsaps College 11 Dr. Yan Wang, Millsaps College James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

42. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix A: Sources 12 http://advat.blogspot.com/2012/04/bayes-factor-analysis- of-extrasensory.html 13 B.S. Sathyaprakash and Bernard F. Schutz, "Physics, Astrophysics and Cosmology with Gravitational Waves", Living Rev. Relativity 12, (2009), 2. URL (cited on May 31, 2013): http://www.livingreviews.org/lrr-2009-2 14 http://www.rzg.mpg.de/visualisation/scientiﬁcdata/projects James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

43. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix B: Probability Theory Key Concepts P(A) ∈ [0, 1] P(Ac) = 1 − P(A) P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A) If A ∩ B = ∅, P(A ∩ B) = P(A)P(B) James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

44. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix B: Probability Theory The Law of Total Probability Consider H = {Hi|i = 1, ..., 6} ⊂ I, where H is mutually exclusive and exhaustive <only 2>P(D) = 6 i=1 P(D ∩ Hi) = 6 i=1 P(D|Hi)P(D) James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

45. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix B: Probability Theory Bayes’ Theorem H = {Hi|i = 1, ..., N} ⊂ I James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

46. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix B: Probability Theory Bayes’ Theorem P(Hi| −→ d , I) = P(Hi|I)P( −→ d |Hi, I) P( −→ d |I) = P(Hi|I)P( −→ d |Hi, I) N i=1 P( −→ d |Hi, I) James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

47. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix B: Probability Theory Bayes’ Theorem P( −→ d |Hi, I)P(Hi|I) = P( −→ d |I)P(Hi| −→ d , I) Likelihood × Prior = Evidence × Posterior L(x) × π(x) = Z × P(x) James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

48. Introduction Data Analysis Nested Sampling Variable Resolution Appendices Appendix A: Slide Sources Appendix B: Probability Theory Appendix C: Nested Sampling Pseudo-Code Appendix C: Nested Sampling Pseudo-Code 1. Draw N points −→ θ a, a ∈ 1...N from prior p( −→ θ ) and calculate their La’s. 2. Set Z0 = 0, i = 0, log(w0) = 0 3. While Lmax wi > Zie−5 a) i = i + 1 b) Lmin = min({La}) c) log(wi ) = log(wi−1) − N−1 d) Zi = Zi−1 + Lminwi e) Replace −→ θ min with −→ θ p( −→ θ |H, I) : L( −→ θ ) > Lmin 4. Add the remaining points: For all a ∈ 1...N, Zi = Zi + L( −→ θ a)wi James Michael Bell Reﬁning Bayesian Data Analysis for Longer Waveforms

Refining Bayesian Data Analysis Methods for Use with Longer Waveforms

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