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Bayesian Inferencewith
physical models of the cosmic Large Scale Structure
Jens JascheGuilhem Lavaux, Florent Leclercq, Alexander Merson,
Benjamin Wandelt
Lightcone Workshop, Garching, 19 October 2016
The cosmic large scale structure...
... A source of knowledge!
- Cosmological parameter
- Geometry of the Universe
- Constituents of the Universe
- Tests of Gravity
- Galaxy formation
- fundamental physics
Image credit: Volker Springel for the Millenium Simulation, MPA Garching
The need for statistics
“If your experiment needs statistics, you ought to do a better experiment.”Lord Ernest Rutherford
No unique recovery! We need statistics!
Inference of signals = Ill-posed problem
Cosmic varianceNoise
Survey geometrySelection effectsForegroundsGalaxy bias
Finite resolution of telescopeRedshift distortionsPhotometric uncertainties
4D Bayesian Inference
Gravity
Initial State Final State
Bayesian physical inference
Statistically complex final state
Statistically simple initial state
Solve inverse Problem via forward modeling
4D Bayesian Inference
Gravity
Initial State Final State
Bayesian physical inference
Statistically complex final state
Statistically simple initial state
Solve inverse Problem via forward modeling
Requires to handle >10 million parameters
( Parameter ) space: The final Frontier
“The Curse of dimensionality” Bellman, 1961
High dimensional parameter spaces are sparse!!!
( Parameter ) space: The final Frontier
“The Curse of dimensionality” Bellman, 1961
High dimensional parameter spaces are sparse!!!
( Parameter ) space: The final Frontier
“The Curse of dimensionality” Bellman, 1961
High dimensional parameter spaces are sparse!!!
( Parameter ) space: The final Frontier
“The Curse of dimensionality” Bellman, 1961
High dimensional parameter spaces are sparse!!!
Bona fide PDFs:
( Parameter ) space: The final Frontier
“The Curse of dimensionality” Bellman, 1961
High dimensional parameter spaces are sparse!!!
Bona fide PDFs:
Space filling and random sampling methods will fail!!!
( Parameter ) space: The final Frontier
“The Curse of dimensionality” Bellman, 1961
High dimensional parameter spaces are sparse!!!
Bona fide PDFs:
Space filling and random sampling methods will fail!!!
But gradients are very valuable!!!
MCMC in high dimensions
HMC: Use Classical mechanics to solve statistical problems!
The potential :
see e.g. Duane et al. (1987)
MCMC in high dimensions
HMC: Use Classical mechanics to solve statistical problems!
The potential :
The Hamiltonian :
see e.g. Duane et al. (1987)
MCMC in high dimensions
HMC: Use Classical mechanics to solve statistical problems!
The potential :
The Hamiltonian :
see e.g. Duane et al. (1987)
Nuisance parameter!!!
MCMC in high dimensions
HMC: Use Classical mechanics to solve statistical problems!
The potential :
The Hamiltonian :
see e.g. Duane et al. (1987)
Nuisance parameter!!!
MCMC in high dimensions
HMC: Use Classical mechanics to solve statistical problems!
The potential :
The Hamiltonian :
see e.g. Duane et al. (1987)
Nuisance parameter!!!
MCMC in high dimensions
HMC: Use Classical mechanics to solve statistical problems!
The potential :
The Hamiltonian :
see e.g. Duane et al. (1987)
Nuisance parameter!!!
HMC beats the “curse of dimensionality” by:
Exploiting gradients
Using conserved quantities
Randomize and accept :
4D Bayesian Inference
BORG
BORG (Bayesian Origin Reconstruction from Galaxies)
Incorporates physical model into Likelihood
Approximate LSS formation model (2LPT / PM)
Uses HMC to solve initial conditions problem
Jasche, Wandelt (2013)
Data application
Analyzing the SDSS DR7 main sample
750 Mpc/h box
~3 Mpc/h grid resolution
Treatment of luminosity dependent bias
Automatic noise calibration
Credit: M. Blanton and the Sloan Digital Sky Survey
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4D Bayesian inference
BORG performs MCMC sampling
Explores space of physical 3D initial conditions
Cartoon! In Reality we deal with 10^7
dimensions!!
4D analysis of the SDSS
Initial density fieldz ~ 1000
final density fieldz=0
SDSS number counts
z=0
Jasche et al. 2015 ( arXiv:1409.6308 )
4D analysis of the SDSS
Dynamic Information
Plausible LSS formation history
Jasche et al. 2015 ( arXiv:1409.6308 )
4D analysis of the SDSS
Jasche et al. 2015 ( arXiv:1409.6308 )
Dynamic Information
Inference of 3D velocity fields
4D Bayesian inference
Lavaux & Jasche 2016 (arXiv:1509.05040 )
4D Bayesian inference
A factor 10 in S/N with respect to state-of-the-art!
Lavaux & Jasche 2016 (arXiv:1509.05040 )
Comparing inference schemesGaussian Log-normal-Poisson 2LPT-Poisson
a.k.a: Wiener-filteringZaroubi et al. 1994Erdogdu et al. 2004Kitaura & Ensslin 2008Grannet et al. 2015
Kitaura 2010Jasche&Kitaura 2010
log-normal-filtering
Jasche&Wandelt 2012
Jasche & Lavaux (in prep)
Which scheme performs best?
Ask the data!
Some thoughts on light cone modeling
Physical modeling of light cone effects in Bayesian inferences
Deep surveys are statistical inhomogeneous
Not a mere scaling of density amplitudes
Speed vs. Accuracy
Are there fast and accurate light cone models?
Summary & Conclusion
4D Bayesian inference
From 3D to 4D (Spatio-Temporal inference)
Non-linear, non-Gaussian
Handling of noise, bias, selection effects, survey geometries
etc.
Scientific results
Physical modeling helps to make the most of data
Characterization of initial conditions
Higher order statistics
Dynamic information, Structure formation
Greatly improved reconstructions of LSS
The end...
Thank You!
−5
0
5 −5
0
50
0.2
0.4
4D Bayesian inference
BORG performs MCMC sampling
Explores space of physical 3D initial conditions
Cartoon! In Reality we deal with 10^7
dimensions!!
−5
0
5 −5
0
50
0.2
0.4
4D Bayesian inference
BORG performs MCMC sampling
Explores space of physical 3D initial conditions
Cartoon! In Reality we deal with 10^7
dimensions!!
Dark matter voids in the SDSS
Leclercq et al. 2015 (arXiv:1410.0355)
Cumulative void number functions
