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

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

Prior and Posterior Probability Likelihood function Prior probability

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 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

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)

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)

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

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

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

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

Thank You ! Junwei Cao jcao@tsinghua.edu.cn http://ligo.org.cn