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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