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
IntroductionData Analysis
Nested SamplingVariable Resolution
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Compact BinariesMotivationResearch Objectives
MotivationTechnology & Limits
Initial LIGO and Virgodetectors
Signal visibility ~30s
Advanced DetectorConfiguration
Signal Visibility >3min
Increased Efficiency⇒ Increased Data Use⇒ More Significant Results
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Compact BinariesMotivationResearch Objectives
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Compact BinariesMotivationResearch Objectives
Research Objectives
PrimaryTo investigate increased parallelization of the existingnested sampling algorithm
SecondaryTo develop a variable resolution algorithm that will improvethe handling of template waveforms
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Compact BinariesMotivationResearch Objectives
Research Objectives
PrimaryTo investigate increased parallelization of the existingnested sampling algorithm
SecondaryTo develop a variable resolution algorithm that will improvethe handling of template waveforms
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
Bayes’ TheoremDerivation
H = {Hi |i = 1, ...,N} ⊂ I and D ⊂ H
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
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)
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
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?
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
Parameter EstimationMarginalization
Goals:
Find the distribution of eachparameter
Find the expectation of eachparameter
Two Parameter Marginalization
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
Model SelectionBayesian 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 Weak3-10 i Substantial
10-30 i Strong30-100 i Very Strong> 100 i Decisive
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Bayes’ TheoremParameter EstimationModel Selection
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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)
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe Procedure
1 Map Θ to R1.
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe 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
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe Procedure
3 Order {xi |i = 1...N} from greatest to least L.
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe Procedure
4 Remove Xj corresponding to Lmin.
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe Procedure
5 Store the smallest sample Xj and its corresponding L(x).
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe Procedure
6 Draw Xi+1 ∈ U(0,Xi) to replace Xi corresponding to Lmin.
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe Procedure
7 Repeat, shrinking {Xi } to regions of increasing likelihood.
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe 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
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe Result
9 Sample from Area Z → Sample from P(x) = L(x)/Z
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
Nested SamplingThe 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
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
The AlgorithmThe ResultParallelization
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
MotivationThe GoalBrainstorming
Variable ResolutionThe 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
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
MotivationThe GoalBrainstorming
Variable ResolutionBrainstorming
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
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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.
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code
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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code
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)
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code
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)
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code
Appendix B: Probability TheoryBayes’ Theorem
H = {Hi |i = 1, ...,N} ⊂ I
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code
Appendix B: Probability TheoryBayes’ 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
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix C: Nested Sampling Pseudo-Code
Appendix B: Probability TheoryBayes’ Theorem
P(−→d |Hi , I)P(Hi |I) = P(
−→d |I)P(Hi |
−→d , I)
Likelihood × Prior = Evidence × PosteriorL(x)× π(x) = Z × P(x)
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms
IntroductionData Analysis
Nested SamplingVariable Resolution
Appendices
Appendix A: Slide SourcesAppendix B: Probability TheoryAppendix 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) = 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
James Michael Bell Refining Bayesian Data Analysis for Longer Waveforms