1. Probability and Statistics
Junwei Cao (曹军威) and Junwei Li (李俊伟)
Tsinghua University
Gravitational Wave Summer School Kunming, China, July 2009

2. 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

3. 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

4. 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

5. Two Approaches of Probability(contd.)
Bayesian approach
Bayes’ theorem

6. Prior and Posterior Probability
Likelihood function
Prior probability

7. 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

8. Before Parameter Estimation
A number of free parameters
A family of possible templates
Denoted generically as
is a collection of parameters
A family of optimal filters
Determined by eq. 2.12,
Must discretize the θ-space
For some template, the SNR exceeds a predefined threshold, indicating a detection

9. Parameter Estimation How to reconstruct parameters of the source
Assume that n(t) is stationary and Gaussian
Corresponding Gaussian probability distribution for n(t)
This is the probability that n(t) has a given realization n0(t)

10. Likelihood Function Assumption
θt is the true value of the parameters θ
Likelihood function for s(t), by plugging n0=s-h(θt) into 3.5
Introduce ht≡h(θt)

11. Posterior Probability Distribution
Introduce a prior probability to eq. 3.7
can be an un-flat prior in θt
Estimator: a rule for assigning , the most probable value of θt
Consistency
The bias
Efficiency
Robustness

12. Maximum Likelihood Estimator
First, the prior probability is flat
Max. posterior  Max. likelihood
Denote the maximum likelihood estimator by
Simpler to maximize
Define

13. 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
Drawback
An ambiguity on the value of the most probable value of θ1
Unable to minimize the error on θ1 determination

14. Bayes Estimator Most probable value of parameters Errors
The “operational” meaning
Is the value of , averaged over an ensemble of same outputs
Drawback: computational costs

15. Thank You !
Junwei Cao