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 Parameter vs Rapid Sky estimation from Markov-Chain Head-to-head Localization Inspirals Monte-Carlo (or just how much Parameter does the Fisher Estimation matrix suck?) Wednesday, May 30, 12
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 Wednesday, May 30, 12
Fisher Information Matrix • Assume gaussian, stationary noise and high signal-to-noise ratio, and define an inner product: Z ∞ a ( f )˜ ˜ b ∗ ( f ) h ab i = 4 < d f S n ( f ) 0 • To first order, � − 1 2 Γ ab ∆ θ a ∆ θ b p ( θ | s ) ∝ p ( θ ) exp ⌧ ∂ h � ∂ h • where and Σ ab = ( Γ − 1 ) ab Γ ab ≡ ∂θ a ∂θ b (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 p ( d | params) • Repeat, constructing probability distribution from Bayes rule: p (params | d ) ∝ p ( d | params) p (params) Wednesday, May 30, 12
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 Wednesday, May 30, 12
MCMC Sampling Wednesday, May 30, 12
MCMC Sampling Wednesday, May 30, 12
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 Wednesday, May 30, 12
So how well does the Fisher matrix really do, compared to a real parameter estimation technique? Wednesday, May 30, 12
MCMC vs Fisher Matrix • Since we assume posterior from MCMC to be correct, normalize Fisher matrix errors by MCMC standard deviations ∆ F IM ∆ MCMC • 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 Wednesday, May 30, 12
MCMC vs Fisher Matrix Wednesday, May 30, 12
MCMC vs Fisher Matrix • Instead of undercutting the MCMC error bars, the Fisher matrix wildly overestimates, up to ~8 times worse • Beginning at ~ 10 M � • For chirp mass, one of the best recovered parameters • Combination of two effects: junk parameter correlations and specific noise realizations Wednesday, May 30, 12
Junk Parameter Correlations Wednesday, May 30, 12
Junk Parameter Correlations Wednesday, May 30, 12
Junk Parameter Correlations • No (good) way to include prior information • But you can kludge it Wednesday, May 30, 12
Junk Parameter Correlations • No (good) way to include prior information • But you can kludge it • With priors, you shrink the overall error ellipsoid volume • Even then, still doesn’t bring it into alignment with MCMC • Need global understanding of parameter space Wednesday, May 30, 12
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 overestimate 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 Wednesday, May 30, 12
MCMC Sky Localization • Why use MCMC? • Template-based, fully coherent search allows for near- optimal 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 Wednesday, May 30, 12
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
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