Parameter Estimation and the Fisher Matrix in Advanced LIGO Carl - - PowerPoint PPT Presentation

Parameter Estimation and the Fisher Matrix in Advanced LIGO

Carl Rodriguez Ilya Mandel Ben Farr, Vivien Raymond, Will Farr, Vicky Kalogera

Wednesday, May 30, 12

Overview

Fisher Matrix vs Markov-Chain Monte-Carlo Parameter Estimation Head-to-head (or just how much does the Fisher matrix suck?) Parameter estimation from Inspirals Rapid Sky Localization

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The Problem

• Advanced LIGO receives a signal:
• Template matching only gets you a rough guess
• Astrophysics requires accurate measurement of system

parameters

• Inspiral pipeline passes trigger and best guess for

parameter estimation

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Fisher Information Matrix

• Assume gaussian, stationary noise and high signal-to-noise ratio, and

define an inner product:

• To first order,
• where and

p(θ|s) ∝ p(θ) exp  −1 2Γab∆θa∆θb

• Γab ≡

⌧ ∂h ∂θa ∂h ∂θb

• habi = 4<

Z ∞ ˜ a(f)˜ b∗(f) Sn(f) d f Σab = (Γ−1)ab (Fisher Matrix)

Wednesday, May 30, 12

Fisher Information Matrix

• What’s it good for?
• 1. Pick a point in parameter space
• 2. The FIM returns gaussian uncertainties about that point
• Cheap, computationally simple
• Theoretical lower limit on parameter uncertainties
• Poor person’s “exploration” of parameter space
• Used entirely too much

Wednesday, May 30, 12

Fisher Information Matrix Pitfalls

• No (good) way to include prior information
• No global exploration of the parameter space
• (Multiple peaks)
• Walls in parameter space
• Highly susceptible to crazy cross correlations
• These problems are frequently ignored...

Wednesday, May 30, 12

Parameter Estimation

• After detection, determining parameters that produced a compact

binary coalescence waveform

• Producing Bayesian probability distributions
• Start with a stretch of data and some guess parameters for the system
• Subtract template from noise and compute
• Repeat, constructing probability distribution from Bayes rule:

p(d|params) p(params|d) ∝ p(d|params)p(params)

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MCMC Sampling

• Still need efficient way to sample 15 dimensional parameter

space

• Use Markov-Chain Monte Carlo:
• 1. propose a jump in parameter space
• 2. accept if signal fit is better
• 3. reject if fit is worse (sometimes)
• “Chains” trace out parameter space efficiently, finding multiple

modes despite parameter degeneracies

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MCMC Sampling

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MCMC Sampling

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Disadvantages of MCMC

• Slow for theoretical exploration of parameter space:
• 1 day on HPC vs 1 second on laptop for Fisher Matrix
• Have to be extremely clever about jump proposals
• Requires knowledge of parameter space

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So how well does the Fisher matrix really do, compared to a real parameter estimation technique?

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MCMC vs Fisher Matrix

• Since we assume posterior from MCMC to be correct,

normalize Fisher matrix errors by MCMC standard deviations

• Run over 200 random signals and compare results
• As Fisher Matrix represents statistical upper bound, expect

most signals for well detected parameters < 1

• Look at Chirp Mass error fraction as function of total mass

∆F IM ∆MCMC

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MCMC vs Fisher Matrix

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MCMC vs Fisher Matrix

• Instead of undercutting the MCMC error bars, the Fisher

matrix wildly overestimates, up to ~8 times worse

• Beginning at ~
• For chirp mass, one of the best recovered parameters
• Combination of two effects: junk parameter correlations

and specific noise realizations 10M

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Junk Parameter Correlations

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Junk Parameter Correlations

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Junk Parameter Correlations

• No (good) way to include

prior information

• But you can kludge it

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Junk Parameter Correlations

• No (good) way to include

prior information

• But you can kludge it
• With priors, you shrink the
• verall error ellipsoid volume
• Even then, still doesn’t bring it

into alignment with MCMC

• Need global understanding of

parameter space

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Conclusions

• Studies done with Fisher matrix were good first step, but quite

unhelpful when compared to real parameter estimation

• You can miss correlations and priors which wildly
• verestimate the errors
• You can miss entire peaks and horribly underestimate the

errors

• In order to know where this happens, you need to

already know about the parameter space (at which point you might as well do MCMC anyway)

Wednesday, May 30, 12

MCMC Sky Localization

• Project by Ben Farr and Northwestern group to use

repurposed MCMC

• Replace triangulation with low latency MCMC run;

several advantages over triangulation

• Currently done via triangulation:
• Extremely fast (~1s)
• Employs galaxy catalogue as prior

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MCMC Sky Localization

• Why use MCMC?
• Template-based, fully coherent search allows for near-
• ptimal sky localization
• Gain additional parameters (distance, inclination, etc.)
• Fix masses, only optimizing over extrinsic parameters
• Only have to recompute antennae function, much faster

than all-parameter MCMC

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MCMC Sky Localization

• MCMC can run on order of

minutes

• No significant loss vs full

MCMC

• Confidence intervals returned

are comparable to those from full MCMC

Ben Farr, APS 2012

Wednesday, May 30, 12