Download PowerPoint Template

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