3.4 Likelihood ratio test

It is remarkable that all three very different approaches – Bayesian, minimax, and Neyman–Pearson – lead to the same test called the likelihood ratio test [34]. The likelihood ratio is the ratio of the pdf when the signal is present to the pdf when it is absent:
We accept the hypothesis if , where is the threshold that is calculated from the costs , priors , or the significance of the test depending on what approach is being used.

3.4.1 Gaussian case – The matched filter

Let be the gravitational-wave signal and let be the detector noise. For convenience we assume that the signal is a continuous function of time and that the noise is a continuous random process. Results for the discrete time data that we have in practice can then be obtained by a suitable sampling of the continuous-in-time expressions. Assuming that the noise is additive the data can be written as

In addition, if the noise is a zero-mean, stationary, and Gaussian random process, the log likelihood function is given by
where the scalar product is defined by
In Equation (22) denotes the real part of a complex expression, the tilde denotes the Fourier transform, the asterisk is complex conjugation, and is the one-sided spectral density of the noise in the detector, which is defined through equation
where E denotes the expectation value.

From the expression (21) we see immediately that the likelihood ratio test consists of correlating the data with the signal that is present in the noise and comparing the correlation to a threshold. Such a correlation is called the matched filter. The matched filter is a linear operation on the data.

An important quantity is the optimal signal-to-noise ratio defined by

We see in the following that determines the probability of detection of the signal. The higher the signal-to-noise ratio the higher the probability of detection.

An interesting property of the matched filter is that it maximizes the signal-to-noise ratio over all linear filters [34]. This property is independent of the probability distribution of the noise.