1. Brief Explanation of MCMC implementation in lalsuite
Hyung Won Lee, Inje University
with Chunglee Kim(KHU&KISTI) and Jeongcho Kim(Inje)
6 Feb. 2015, KAGRA f2f meeting, Hongo Campus
11th KAGRA f2f Meeting, 4-7 February 2015

2. 11th KAGRA f2f Meeting, 4-7 February 2015
Contents
Motivation
CBC gravitational wave sources
Parameter Estimation with Bayesian Inference
Implementation in lalsuite
Suggestions for KAGALI
Discussions
11th KAGRA f2f Meeting, 4-7 February 2015

3. 11th KAGRA f2f Meeting, 4-7 February 2015
Motivation
How does it work parameter estimation
Understanding Bayesian inference/MCMC
Which physical parameters
How it was implemented in lalsuite
How it could be implemented for KAGALI
Get practical work out items
11th KAGRA f2f Meeting, 4-7 February 2015

4. 11th KAGRA f2f Meeting, 4-7 February 2015
CBC GW sources
Neutron star(NS) + Black hole(BH) binary
Mass range : 1 𝑀 ⨀ ~ 𝑀 ⨀
Frequency range : 1𝐻𝑧 ~ 10𝑘𝐻𝑧
Waveform : Inspiral, Merger, Ringdown
11th KAGRA f2f Meeting, 4-7 February 2015

5. CBC Inspiral GW Waveform
11th KAGRA f2f Meeting, 4-7 February 2015

6. 11th KAGRA f2f Meeting, 4-7 February 2015
Time Domain Template
TaylorT1, 2, 3, 4
PadeT1 : P-approximation
EOB : effective one body
SpinTaylorT1, 2, 3, 4, 5
IMRPhenomA, B, C
TaylorT2 can deal with CBC inspirals when there is a BH spin that is aligned to the orbital angular momentum.
SpinTaylorT4 is one of the most comprehensive inspiral waveform model used by the LIGO-Virgo collaboration. This can deal with precessing CBC inspirals with two spins.
11th KAGRA f2f Meeting, 4-7 February 2015

7. Frequency Domain Template
TaylorF1
TaylorF2 : standard for detection pipeline, SPA approximation
TaylorF2Amp : include higher harmonics due to amplitude corrections up to 2.5 pN
PadeF1
BCV
BCVSpin
11th KAGRA f2f Meeting, 4-7 February 2015

8. 11th KAGRA f2f Meeting, 4-7 February 2015
TaylorF2Amp Waveform
Arun et.al., PRD79, (2009)
Amplitude corrections up to n=5 (2.5pN)
11th KAGRA f2f Meeting, 4-7 February 2015

9. 11th KAGRA f2f Meeting, 4-7 February 2015
SPA phase factor +
Phase corrections up to 3.5PN (standard in LAL)
11th KAGRA f2f Meeting, 4-7 February 2015

10. SPA phase factor(Code)
11th KAGRA f2f Meeting, 4-7 February 2015

11. 11th KAGRA f2f Meeting, 4-7 February 2015
Physical Parameters
Individual masses
Luminosity distance
Inclination angle
Coalescence phase
Coalescence time at geocenter
Declination
Right ascension
Polarization
Spins, quadrupole deformations
11th KAGRA f2f Meeting, 4-7 February 2015

12. Parameter Estimation with Bayesian inference
℘ 𝜃 𝑥 = ℘ 𝜃, 𝑥 ℘ 𝑥 = ℘(𝑥|𝜃)℘ 𝜃 ℘(𝑥|𝜃)℘ 𝜃 𝑑𝜃
℘ 𝜃,𝑥 =℘ 𝑥 𝜃 ℘ 𝜃 =℘ 𝜃 𝑥 ℘ 𝑥
𝜃 : unobservable model parameters
𝑥 : observable data
℘ 𝜃,𝑥 : Joint probability observing data 𝑥 with model parameter 𝜃
℘ 𝑥 𝜃 ∝𝐿𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
Metropolis-Hasting Algorithm, ~ samples
11th KAGRA f2f Meeting, 4-7 February 2015

13. Parameter Estimation with Bayesian inference
MCMC Samples
PTMCMCOutput.00
PTMCMCOutput.nn
Post Process
cbcByesPostProc.py
11th KAGRA f2f Meeting, 4-7 February 2015

14. 11th KAGRA f2f Meeting, 4-7 February 2015
Overlap Calculation
Template
Data
𝜌=4ℜ 0 +∞ ℎ (𝑓) 𝑠 ∗ (𝑓) 𝑆 𝑛 (𝑓) 𝑑𝑓
𝑆𝑁𝑅 2 =4 0 +∞ ℎ (𝑓) 2 𝑆 𝑛 (𝑓) 𝑑𝑓
Network SNR 𝑆𝑁𝑅 2 = 𝑖=1 𝑁 𝑆𝑁𝑅 𝑖 2
One-sided power spectral density of noise
11th KAGRA f2f Meeting, 4-7 February 2015

15. Likelihood Calculation
ℒ(𝑠|𝜃)∝exp −2 0 ∞ 𝑠 𝑓 − ℎ (𝜃,𝑓) 2 𝑆 𝑛 (𝑓) 𝑑𝑓
11th KAGRA f2f Meeting, 4-7 February 2015

16. 11th KAGRA f2f Meeting, 4-7 February 2015
Various Overlaps
Unnormalized
𝜌=4ℜ 0 +∞ ℎ 1 (𝑓) ℎ 2 ∗ (𝑓) 𝑆 𝑛 (𝑓) 𝑑𝑓
Normalized
𝜌 𝑘 =4ℜ 0 +∞ ℎ 𝑘 (𝑓) ℎ 𝑘 ∗ (𝑓) 𝑆 𝑛 (𝑓) 𝑑𝑓
𝜌 𝑁 = 𝜌 𝜌 1 𝜌 2
11th KAGRA f2f Meeting, 4-7 February 2015

17. 11th KAGRA f2f Meeting, 4-7 February 2015
MCMC Process
Accept next state with probability
𝑟= ℘(𝑥| 𝜃 2 ) 1/𝑇 ℘ 𝜃 ℘(𝑥| 𝜃 1 ) 1/𝑇 ℘ 𝜃 1
Metropolis-Hasting smapling
11th KAGRA f2f Meeting, 4-7 February 2015

18. 11th KAGRA f2f Meeting, 4-7 February 2015
Parallel Tempering
Use few chains with different temperature
Use likelihood ℒ 𝑠 𝜃 1 𝑇 , 𝑇>1
Improve convergence
Improve mixing
The higher temperature, the smoother distribution
𝑇>1
11th KAGRA f2f Meeting, 4-7 February 2015

19. Implementation in lalsuite
LAL MCMC Pipeline
Directory structure of lalsuite
Basic entities of LALSuite
LAL applications
Likelihood, proposal and prior functions
Simulation related functions and waveforms
Post Processing python script
11th KAGRA f2f Meeting, 4-7 February 2015

20. 11th KAGRA f2f Meeting, 4-7 February 2015
LAL MCMC Pipeline
Start
Initialization
MPI synchronize
PT MCMC
Clearance
End
11th KAGRA f2f Meeting, 4-7 February 2015

21. LALInferenceRunState
ProcessParamsTable *commandLine; /** A ProcessParamsTable with command line arguments */
LALInferenceInitModelFunction initModel; /** A function that returns a new set of variables for the model */
LALInferenceAlgorithm algorithm; /** The algorithm function */
LALInferenceEvolveOneStepFunction evolve; /** The algorithm's single iteration function */
LALInferencePriorFunction prior; /** The prior for the parameters */
LALInferenceCubeToPriorFunction CubeToPrior; /** MultiNest prior for the parameters */
LALInferenceLikelihoodFunction likelihood; /** The likelihood function */
LALInferenceProposalFunction proposal; /** The proposal function */
LALInferenceLogFunction logsample; /** Log sample, i.e. to disk */
LALInferenceTemplateFunction templt; /** The template generation function */
… more
11th KAGRA f2f Meeting, 4-7 February 2015

22. LAL MCMC pipeline in a glance
MCMC main
main
(start)
parseCommandLine Options
Initialize Rusnstate
initializeMCMC
addVariables
InitializeCBC
Initialize likelihood
Initialize MCMC state
MPI_Barrier wait synchronization
runState ->alogorithm
main MCMC routine
PTMCMCAlgorithm
Deinitialize
Clean resources
end
11th KAGRA f2f Meeting, 4-7 February 2015

23. PTMCMCAlgorithm No Yes Receive runComplete state from root process
(start)
PTMCMCAlgorithm
MPI setup, runComplete = 0
Ladder setup for parallel tempering Ladder[i] = Min ( 𝑀𝑎𝑥 𝑀𝑖𝑛 ) 𝑖 𝑛𝐶ℎ𝑎𝑖𝑛−1
Set temperature for this chain
if(MPI_rank == 0) output
Anealing related setup
Debug related setup
Receive runComplete state from root process
If(MPI_rank≠0) non-blocking receive
MPI_Barrier No
!runComplete
Yes
Ladder update if need
Calculate ACL(autocorrelation length) If( i>Neff) runComplete=1
Anealing
post-anealing
runState ->evolve
OneStep MCMC routine
PTMCMCOneStep
Save for every Nskip step
If(i>Niter) runComplete=1
If(MPI_rank==0 && runComplete==1) send run complete value to all other parallel process
MPI_Barrier
XLALFree() resource free
end
11th KAGRA f2f Meeting, 4-7 February 2015

24. LALInferenceCyclicProposal
PTMCMCOneStep
(start)
PTMCMCOneStep
MPI setup
runState ->proposal
LALInferenceCyclicProposal
Compute prior & likelihood
determine acceptance probability
Accept
end
LALInferenceCyclicProposal
LALInferenceCyclicProposal
(start)
LALInferenceProposalFunction *cycle = NULL
Must have cycle array and cycle array length in propArgs
(cycle[i])(runState, proposedParams)
Call proposal
end
11th KAGRA f2f Meeting, 4-7 February 2015

25. LALInferenceUndecomposed FreqDomainLogLikelihood()
Mcmc main
PTMCMCAlgorithm()
PTMCMCOneStep()
LALInferenceUndecomposed
FreqDomainLogLikelihood()
LALInferenceTemplate
XLALSimInspiralChooseWaveform()
XLALSimInspiralChooseFD
WaveformFromCache()
XLALSimInspiralChooseFDWaveform()
XLALSimInspiralTaylorF2Amp()
Call TaylorF2Amp in MCMC
11th KAGRA f2f Meeting, 4-7 February 2015

26. LAL parameter estimation(PE) pipeline
1. waveform library [LALSimulation]
2. PE library [LALInference & LALapps]
3. basic libraries are scattered, but most basic ones are in [LAL]
11th KAGRA f2f Meeting, 4-7 February 2015

27. waveform library [LALSimulation]
LALSimulation/src/ ..
LALSimInspiral.h -> protocol definitions
XLALSimInspiralF2AmpPlus() -> ℎ +
XLALSimInspiralF2AmpCross() -> ℎ ×
LALSimInspiral.c -> XLALSimInspiralChooseFDWaveform() call a waveform model
XLALSimInspiralF2AmpPlus()
XLALSimInspiralF2AmpCross()
LALSimInspiralTaylorF2Amp.c -> example of a FD waveform
LALSimInspiralWaveformCache.c -> XLALSimInspiralChooseFDWaveformFromCache()
call XLALSimInspitalChooseFDWaveform()
11th KAGRA f2f Meeting, 4-7 February 2015

28. PE library [LALInference]
LALInference/src/..
LALInference.h
LALInferenceReadData.c
LALInferecneTemplate.c
LALInference.h -> collections of structures
typedef tagLALInferenceRunState signal/parameters {
∙∙∙ -> Structure to contain inference run state
} LALInferenceRunState;
typedef struct tagLALInferenceIFOData data for detector including sensitivity
∙∙∙ -> Structure to contain IFOdata
} LALInferenceIFOData;
11th KAGRA f2f Meeting, 4-7 February 2015

29. PE library [LALInference]
LALInference/src/..
LALInference.h
LALInferenceReadData.c
LALInferecneTemplate.c
reads injection parameters from an xml file
reads PSD for each detector
reads data for each detector
generate noise for each detector
LALInferenceIFOData *LALInferenceReadData(ProcessParamsTable *commandLine)
-> Read in the data and store it in a LALInferenceIFOData structure
11th KAGRA f2f Meeting, 4-7 February 2015

30. PE library [LALInference]
LALInference/src/..
LALInference.h
LALInferenceReadData.c
LALInferecneTemplate.c
- Jump around parameter space 𝜆 𝑗 generate template signal
LALInferenceTemplate.c -> template calls to LAL template functions
LALInferenceTemplateXLALSimInspiralChooseWaveform() Wrapper for LALSimulation waveforms : XLALSimInspiralChooseFDWaveform()
XLALSimInspiralChooseTDWaveform()
- Parameters of non-spinning waveform in LALInferencePrior.c
11th KAGRA f2f Meeting, 4-7 February 2015

31. PE library [LALInference]
LALInference/src/..
LALInference.h
LALInferenceLikelihood.c
LALInferecneTemplate.c
LALInferenceLikelihood.c -> Bayesian Followup likelihood function
LALInferenceUndeomposedFreqDomainLogLikelihood()
Required (`currentParams') parameters are:
"rightascension" (REAL8, radian, 0 <= RA <= 2pi)
"declination" (REAL8, radian, -pi/2 <= dec <=pi/2)
"polarisation” (REAL8, radian, 0 <= psi <= pi)
"time" (REAL8, GPS sec.)
11th KAGRA f2f Meeting, 4-7 February 2015

32. 11th KAGRA f2f Meeting, 4-7 February 2015
PE library [LALapps]
LALapps/src/inspiral/posterior/..
LALInferenceMCMC.c
LALInferenceMCMCSampler.c
LALInferenceMCMC.c -> Bayesian Followup function testing site
main()
1. Set up structures for MCMC
2. Choose the likelihood function -> ex) student-T , Undecomposed.
3. Call MCMC algorithm -> sampling posterior function
11th KAGRA f2f Meeting, 4-7 February 2015

33. 11th KAGRA f2f Meeting, 4-7 February 2015
PE library [LALapps]
LALapps/src/inspiral/posterior/..
LALInferenceMCMC.c
LALInferenceMCMCSampler.c
LALInferenceMCMCSampler.c -> Bayesian Followup, MCMC algorithm
PTMCMCAlgorithm()
PTMCMCOneStep() -> Metropolis-Hasting algorithm implemented
11th KAGRA f2f Meeting, 4-7 February 2015

34. basic functions, libraries
lal/FrequencySeries.h -> dataStructure
lal/LALDatatypes.h ->Primitive datatypes
Aggregate datatypes
Store arbitrarily large sets or collections of primitive datatypes.
aggregate datatypes are defines
Structured datatypes : embed primitive and aggregate datatypes
inside structures that define their physical meaning.
lal/LALConstants.h -> physical constants, Pi, h, G, etc.
11th KAGRA f2f Meeting, 4-7 February 2015

35. Suggestions for KAGALI
Directory structure by functionality
MCMC chain generation
Waveform generation
Input output functions
Notes to consider
Waveform function call parameter
Post process language
Python
Haskell
C and gnuplot
11th KAGRA f2f Meeting, 4-7 February 2015

36. 11th KAGRA f2f Meeting, 4-7 February 2015
Discussions
Bayesian Parameter Estimation
Optimal library structure
How to post process?
All from scratch?
Collaboration meeting from 9 to 11 February at Inje University
11th KAGRA f2f Meeting, 4-7 February 2015