Refining Bayesian Data Analysis Methods for Use with Longer Waveforms

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IntroductionData Analysis

Nested SamplingVariable Resolution

Appendices

Refining Bayesian Data Analysis Methods foruse 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

Appendices

Compact BinariesMotivationResearch Objectives

Coalescing Compact BinariesBlack Hole and Neutron Star Pairs

Primary candidate forground-based GWdetectors.

Expected rate ofoccurrence (per Mpc3Myr )

NS-NS: 0.01 to 10NS-BH: 4× 10−4 to 1BH-BH: 1× 10−4 to 0.3

Implications of findingsFurther validation ofgeneral relativityInsight about physicalextrema

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MotivationTechnology & Limits

Initial LIGO and Virgodetectors

Signal visibility ~30s

Advanced DetectorConfiguration

Signal Visibility >3min

Increased Efficiency⇒ Increased Data Use⇒ More Significant Results

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MotivationRecent Progress in the Field

"Nested Sampling" (2004)J. Skilling

Bayesian coherent analysis of in-spiral gravitational wavesignals with a detector network (2010)

J. Veitch and A. Vecchio

Appendices

Research Objectives

PrimaryTo investigate increased parallelization of the existingnested sampling algorithm

SecondaryTo develop a variable resolution algorithm that will improvethe handling of template waveforms

Appendices

Research Objectives

PrimaryTo investigate increased parallelization of the existingnested sampling algorithm

SecondaryTo develop a variable resolution algorithm that will improvethe handling of template waveforms

Appendices

Bayes’ TheoremParameter EstimationModel Selection

Bayes’ TheoremDerivation

H = {Hi |i = 1, ...,N} ⊂ I and D ⊂ H

Appendices

Bayes’ TheoremDerivation

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)

Appendices

Parameter EstimationIdentifying the parameters

Hypothesis depends on a minimum of 9 parametersΘ = {M, ν, t0, φ0,DL, α, δ, ψ, ι}2 masses, time, sky position, distance, 3 orientation angles

Other possible parameters2 magnitudes and 4 orientation angles for spins2 parameters for the equation of stateMore?

Appendices

Parameter EstimationMarginalization

Goals:

Find the distribution of eachparameter

Find the expectation of eachparameter

Two Parameter Marginalization

Appendices

Parameter EstimationMarginalization 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

Appendices

p(−→θ A|−→d ,H, I) =

∫ΘB

p(−→θ A|−→d ,H, I)d

−→θ B

〈−→θ A〉 =

∫ΘA

−→θ Ap(

−→θ A|−→d ,H, I)d

−→θ A

Appendices

p(−→θ A|−→d ,H, I) =

∫ΘB

p(−→θ A|−→d ,H, I)d

−→θ B

〈−→θ A〉 =

∫ΘA

−→θ Ap(

−→θ A|−→d ,H, I)d

−→θ A

Appendices

Model SelectionBayesian Hypothesis Testing

The Bayes Factor

P(Hj |I)P(−→d |Hj , I)

=P(Hi |I)P(Hj |I)

K H Support Strength< 1 j ?1-3 i Weak3-10 i Substantial

10-30 i Strong30-100 i Very Strong> 100 i Decisive

Appendices

Model SelectionQuantifying 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 ProblemsDimensionality of ΘLarge intervals to integrate

Appendices

Model SelectionQuantifying 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 ProblemsDimensionality of ΘLarge intervals to integrate

Appendices

The AlgorithmThe ResultParallelization

Nested SamplingThe objective of nested sampling

What we need:To calculate the evidence integral using random sample

What we want:To reduce time of evidence computationsTo produce marginalized PDFs and expectationsTo increase accuracy of previous algorithms

Appendices

Nested SamplingBayes’ 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)

Appendices

Nested SamplingThe Procedure

1 Map Θ to R1.

Appendices

2 Draw N samples {Xi |i = 1...N} from π(x) and find L(x).

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3 Order {xi |i = 1...N} from greatest to least L.

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4 Remove Xj corresponding to Lmin.

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5 Store the smallest sample Xj and its corresponding L(x).

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6 Draw Xi+1 ∈ U(0,Xi) to replace Xi corresponding to Lmin.

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7 Repeat, shrinking {Xi } to regions of increasing likelihood.

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Nested SamplingThe Result

8 Area Z =∑1

0 L(x)δx ≈∫ 1

0 L(x)dx shown in (a).

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9 Sample from Area Z → Sample from P(x) = L(x)/Z

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10 Sample from P(x) = L(x)/Z ⇒ Sample from P(−→x |−→d , I)

Appendices

Nested SamplingParallelization of the Existing Algorithm

Run algorithm in parallel with different random seedsSave each sample set and its likelihood values

Collate the results of the multiple runsSort the resulting samples by their likelihood values

Treat samples as part of a collection {NT} =∑Nruns

k=1 NkEach parallel run contains Nk live points

Re-apply nested sampling with lower sample weight

Appendices

MotivationThe GoalBrainstorming

Variable ResolutionMotivation

Improved efficiency with better template waveform handling

Higher resolution⇒ Increased computation timeLower resolution⇒ Decreased accuracy

Current algorithm utilizes stationary resolution function

Appendices

Variable ResolutionThe Goal

Implement a variable resolution function

Exploit the monochromatic nature of the early waveform

Focus computational resources on more complex regions

Appendices

Variable ResolutionBrainstorming

Possible Methods

Time-series variation of least-squares parameters

Event triggering

Monte Carlo Methods and/or further nested sampling

Appendices

Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code

Appendices

Appendix A: Sources

1 arXiv:0911.3820v2 [astro-ph.CO]2 Data Analysis: A Bayesian Tutorial; D.S. Sivia with J.

Skilling3 http://www.stat.duke.edu/~fab2/nested_sampling_talk.pdf4 http://www.mrao.cam.ac.uk/ steve/malta2009/images/

nestposter.pdf5 http://ba.stat.cmu.edu/journal/2006/vol01/issue04/

skilling.pdf6 Dr. John Veitch and Dr. Chris Van Den Broeck

Appendices

Appendix A: Sources

7 http://www.inference.phy.cam.ac.uk/bayesys/8 http://arxiv.org/pdf/0704.3704.pdf9 Dr. Shadow J.Q. Robinson, Millsaps College

10 Dr. Mark Lynch, Millsaps College11 Dr. Yan Wang, Millsaps College

Appendices

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

Appendices

Appendix B: Probability TheoryKey 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)

Appendices

Appendix B: Probability TheoryThe Law of Total Probability

Consider H = {Hi |i = 1, ...,6} ⊂ I, where H is mutuallyexclusive and exhaustive

<only 2>P(D) =∑6

i=1 P(D ∩Hi) =∑6

i=1 P(D|Hi)P(D)

Appendices

Appendix B: Probability TheoryBayes’ Theorem

H = {Hi |i = 1, ...,N} ⊂ I

Appendices

P(Hi |−→d , I) =

P(−→d |I)

=P(Hi |I)P(

−→d |Hi , I)∑N

i=1 P(−→d |Hi , I)

Appendices

P(−→d |Hi , I)P(Hi |I) = P(

−→d |I)P(Hi |

−→d , I)

Likelihood × Prior = Evidence × PosteriorL(x)× π(x) = Z × P(x)

Appendices

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) = 03. While Lmaxwi > Zie−5

a) i = i + 1b) 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

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Appendices

Pseudo-Code

−→θ p(−→θ |H, I) : L(

−→θ ) > Lmin

−→θ a)wi

Refining Bayesian Data Analysis Methods for Use with Longer Waveforms

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