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4.3 Maximum likelihood estimation

Often we do not know the a priori probability density of a given parameter and we simply assign to it a uniform probability. In such a case maximization of the a posteriori probability is equivalent to maximization of the probability density p(x,h) treated as a function of h . We call the function l(h,x) := p(x,h) the likelihood function and the value of the parameter h that maximizes l(h,x) the maximum likelihood (ML) estimator. Instead of the function l we can use the function /\(h, x) = l(h,x)/p(x) (assuming that p(x) > 0). /\ is then equivalent to the likelihood ratio [see Equation (17View Equation)] when the parameters of the signal are known. Then the ML estimators are obtained by solving the equation
@ log-/\(h,x)-= 0, (29) @h
which is called the ML equation .

4.3.1 Gaussian case

For the general gravitational-wave signal defined in Equation (12View Equation) the log likelihood function is given by

T 1- T log /\ = a .N - 2 a .M .a, (30)
where the components of the column vector N and the matrix M are given by
N (k) := (x|h(k)), M(k)(l) := (h(k)| h(l)), (31)
with x(t) = n(t) + h(t), and where n(t) is a zero-mean Gaussian random process. The ML equations for the extrinsic parameters a can be solved explicitly and their ML estimators ^a are given by
^a = M -1 .N. (32)
Substituting ^a into log /\ we obtain a function
1- T - 1 F = 2 N .M .N, (33)
that we call the F -statistic. The F -statistic depends (nonlinearly) only on the intrinsic parameters qm .

Thus the procedure to detect the signal and estimate its parameters consists of two parts. The first part is to find the (local) maxima of the F -statistic in the intrinsic parameter space. The ML estimators of the intrinsic parameters are those for which the F -statistic attains a maximum. The second part is to calculate the estimators of the extrinsic parameters from the analytic formula (32View Equation), where the matrix M and the correlations N are calculated for the intrinsic parameters equal to their ML estimators obtained from the first part of the analysis. We call this procedure the maximum likelihood detection . See Section  4.8 for a discussion of the algorithms to find the (local) maxima of the F -statistic.


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