Maximum likelihood estimation

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 (19

)] when the parameters of the signal are known. Then the ML estimators are obtained by solving the equation

which is called the ML equation.

4.3.1 Gaussian case

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

where the components of the column vector $N$ and the matrix $M$ are given by

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

Substituting $ˆa$ into $log Λ$ we obtain a function

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 (34 ), 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.