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Gravitational Wave Data Analysis Probability and Statistics Junwei Cao (曹军威) and Junwei Li (李俊伟) Tsinghua University Gravitational Wave Summer School Kunming, China, July 2009 Drawback of Matched Filtering Two premises of matched filtering Premise 1: A given signal is present in data stream Premise 2: The form of h(t) is known Premises are impractical Repeat matched filtering with many different filters A number of “events” are extracted “events” indicate that in the detector happened something, which deserves further scrutiny Definition of Probability Consider a set S with subsets A, B… Define probability P as a real function For every A in S, P(A)≥0 For disjoint subsets (i.e. A∩B= ), P(A∪B)=P(A)+P(B) P(S)=1 Further more, conditional probability P( A B) P( A | B) P( B) Two Approaches of Probability Frequentist (also called classical) A, B, … are the outcome of a repeatable experiment, and P(A) is defined as the frequency of occurrence of A In considering the conditional probability, such as P(data | hypothesis), one is never allowed to think about the probability that the parameters take a given a value, nor of the probability that a hypothesis is correct Two Approaches of Probability(contd.) Bayesian approach Bayes’ theorem P( B | A) P( A) P( A | B) P( B) (3.1) P( B) P( B | Ai ) P( Ai ) (3.2) i P( A | B) P( B | A) P( A) P( B | A ) P( A ) i i i (3.3) Prior and Posterior Probability P(hypothesis | data) P(data | hypothesis) P( hypothesis) Posterior probability Likelihood function Prior probability (3.4) Which One to be Chosen Depend on the type of experiment Elementary particle physics is suited for the classical approach since it is the physicist that controls the parameters of the experiment In astrophysics, the sources can be rare, and each one is very interesting individually, e.g. a single BH-BH binary coalesces Before Parameter Estimation A number of free parameters A family of possible templates Denoted generically as h(t ; ) 1,..., N is a collection of parameters A family of optimal filters Denoted generically as K (t; ) ~ ~ Determined by eq. 2.12, K ( f ; ) ~ h( f ; ) / Sn ( f ) Must discretize the θ-space For some template, the SNR exceeds a predefined threshold, indicating a detection Parameter Estimation How to reconstruct parameters of the source Assume that n(t) is stationary and Gaussian Corresponding Gaussian probability distribution for n(t) p(n0 ) N exp (n0 | n0 ) / 2 (3.5) This is the probability that n(t) has a given realization n0(t) Likelihood Function Assumption s(t ) h(t;t ) n0 (t ) θt is the true value of the parameters θ Likelihood function for s(t), by plugging n0=s-h(θt) into 3.5 1 ( s | t ) N exp ( s h(t ) | s h(t )) 2 (3.6) Introduce ht≡h(θt) 1 1 ( s | t ) N exp (ht | s ) (ht | ht ) ( s | s ) 2 2 (3.7) Posterior Probability Distribution (t ) to Introduce a prior probability eq. 3.7 1 p(t | s ) Np (t ) exp ( ht | s) (ht | ht ) 2 (0) p (t ) can be an un-flat prior in θt (0) (3.8) Estimator: a rule for assigning , the most probable value of θt Consistency The bias b E ( ) t Efficiency Robustness Maximum Likelihood Estimator First, the prior probability is flat Max. posterior Max. likelihood (s | t ) Denote the maximum likelihood estimator by ML ( s) Simpler to maximize log 1 log ( s | t ) (ht | s) (ht | ht ) 2 (3.9) Define i / ti (i ht | s) (i ht | ht ) 0 (3.10) Maximum Posterior Probability Maximize the full posterior probability Takes into account the prior probability distribution Non-trivial prior information Example Two-dimensional parameters space (θ1,θ2) ~ Only interested in θ1 p(1 | s) p(1 , 2 | s) Drawback An ambiguity on the value of the most probable value of θ1 Unable to minimize the error on θ1 determination Bayes Estimator Most probable value of parameters Bi ( s) i p( | s)d (3.11) Errors i i j j B B (s) B (s) p( | s)d ij (3.12) The “operational” meaning i i ( s ) B Is the value of , averaged over an ensemble of same outputs Drawback: computational costs Gravitational Wave Data Analysis Junwei Cao [email protected] http://ligo.org.cn