Go to previous page Go up Go to next page

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,θ) treated as a function of θ. We call the function l(θ,x) := p(x,θ) the likelihood function and the value of the parameter θ that maximizes l(θ,x) the maximum likelihood (ML) estimator. Instead of the function l we can use the function Λ (θ,x) = l(θ,x)∕p(x) (assuming that p (x ) > 0). Λ is then equivalent to the likelihood ratio [see Equation (19View Equation)] when the parameters of the signal are known. Then the ML estimators are obtained by solving the equation
∂ log-Λ(θ,x)-= 0, (31 ) ∂ θ
which is called the ML equation.

4.3.1 Gaussian case

For the general gravitational-wave signal defined in Equation (14View Equation) the log likelihood function is given by

T 1- T log Λ = a ⋅ N − 2 a ⋅ M ⋅ a, (32 )
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)), (33 )
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 ˆa are given by
ˆa = M −1 ⋅ N. (34 )
Substituting ˆa into log Λ we obtain a function
1- T − 1 ℱ = 2 N ⋅ M ⋅ N, (35 )
that we call the ℱ-statistic. The ℱ-statistic depends (nonlinearly) only on the intrinsic parameters ξμ.

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 (34View 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.

  Go to previous page Go up Go to next page