Likelihood ratio test

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 $Λ$ is the ratio of the pdf when the signal is present to the pdf when it is absent:

We accept the hypothesis $H1$ if $Λ > k$ , where $k$ is the threshold that is calculated from the costs $Cij$ , priors $πi$ , or the significance of the test depending on what approach is being used.

3.4.1 Gaussian case – The matched filter

Let $h$ be the gravitational-wave signal and let $n$ be the detector noise. For convenience we assume that the signal $h$ is a continuous function of time $t$ and that the noise $n$ 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 $x$ 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

) $ℜ$ denotes the real part of a complex expression, the tilde denotes the Fourier transform, the asterisk is complex conjugation, and $˜S$ 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 $x$ with the signal $h$ 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 $ρ$ defined by

We see in the following that $ρ$ 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.