Fisher information

4.4 Fisher information

It is important to know how good our estimators are. We would like our estimator to have as small variance as possible. There is a useful lower bound on variances of the parameter estimators called Cramèr–Rao bound. Let us first introduce the Fisher information matrix $Γ$ with the components defined by

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The Cramèr–Rao bound states that for unbiased estimators the covariance matrix of the estimators $−1 C ≥ Γ$ . (The inequality $A ≥ B$ for matrices means that the matrix $A − B$ is nonnegative definite.)

A very important property of the ML estimators is that asymptotically (i.e., for a signal-to-noise ratio tending to infinity) they are (i) unbiased, and (ii) they have a Gaussian distribution with covariance matrix equal to the inverse of the Fisher information matrix.

4.4.1 Gaussian case

In the case of Gaussian noise the components of the Fisher matrix are given by

For the case of the general gravitational-wave signal defined in Equation (14

) the set of the signal parameters $θ$ splits naturally into extrinsic and intrinsic parameters: $θ = (a(k),ξμ)$ . Then the Fisher matrix can be written in terms of block matrices for these two sets of parameters as

where the top left block corresponds to the extrinsic parameters, the bottom right block corresponds to the intrinsic parameters, the superscript T denotes here transposition over the extrinsic parameter indices, and the dot stands for the matrix multiplication with respect to these parameters. Matrix $M$ is given by Equation (33

), and the matrices $F$ and $S$ are defined as follows:

The covariance matrix $C$ , which approximates the expected covariances of the ML parameter estimators, is defined as $Γ −1$ . Using the standard formula for the inverse of a block matrix [67 ] we have

where

We call $¯ μν Γ$ (the Schur complement of $M$ ) the projected Fisher matrix (onto the space of intrinsic parameters). Because the projected Fisher matrix is the inverse of the intrinsic-parameter submatrix of the covariance matrix $C$ , it expresses the information available about the intrinsic parameters that takes into account the correlations with the extrinsic parameters. Note that $¯Γ μν$ is still a function of the putative extrinsic parameters.

We next define the normalized projected Fisher matrix

where $√ -T-------- ρ = a ⋅ M ⋅ a$ is the signal-to-noise ratio. From the Rayleigh principle [67

] follows that the minimum value of the component $¯Γ μν n$ is given by the smallest eigenvalue (taken with respect to the extrinsic parameters) of the matrix $((S − FT ⋅ M − 1 ⋅ F) ⋅ M −1)μν$ . Similarly, the maximum value of the component $¯μν Γn$ is given by the largest eigenvalue of that matrix. Because the trace of a matrix is equal to the sum of its eigenvalues, the matrix

where the trace is taken over the extrinsic-parameter indices, expresses the information available about the intrinsic parameters, averaged over the possible values of the extrinsic parameters. Note that the factor 1/4 is specific to the case of four extrinsic parameters. We call $˜Γ μν$ the reduced Fisher matrix. This matrix is a function of the intrinsic parameters alone. We see that the reduced Fisher matrix plays a key role in the signal processing theory that we review here. It is used in the calculation of the threshold for statistically significant detection and in the formula for the number of templates needed to do a given search.

For the case of the signal

the normalized projected Fisher matrix $¯Γ n$ is independent of the extrinsic parameters $A0$ and $φ0$ , and it is equal to the reduced matrix $˜Γ$ [74

]. The components of $˜Γ$ are given by

where $Γ i0j$ is the Fisher matrix for the signal $g(t;ξμ) cos(φ(t;ξμ) − φ0)$ .

Fisher matrix has been extensively used to assess the accuracy of estimation of astrophysically interesting parameters of gravitational-wave signals. First calculations of Fisher matrix concerned gravitational-wave signals from inspiralling binaries in quadrupole approximation [40 , 58 ] and from quasi-normal modes of Kerr black hole [38 ]. Cutler and Flanagan [32 ] initiated the study of the implications of higher PN order phasing formula as applied to the parameter estimation of inspiralling binaries. They used the 1.5PN phasing formula to investigate the problem of parameter estimation, both for spinning and non-spinning binaries, and examined the effect of the spin-orbit coupling on the estimation of parameters. The effect of the 2PN phasing formula was analyzed independently by Poisson and Will [76 ] and Królak, Kokkotas and Schäfer [57 ]. In both of these works the focus was to understand the new spin-spin coupling term appearing at the 2PN order when the spins were aligned perpendicular to the orbital plane. Compared to [57 ], [76 ] also included a priori information about the magnitude of the spin parameters, which then leads to a reduction in the rms errors in the estimation of mass parameters. The case of 3.5PN phasing formula was studied in detail by Arun et al. [12 ]. Inclusion of 3.5PN effects leads to an improved estimate of the binary parameters. Improvements are relatively smaller for lighter binaries.

Various authors have investigated the accuracy with which LISA detector can determine binary parameters including spin effects. Cutler [30 ] determined LISA’s angular resolution and evaluated the errors of the binary masses and distance considering spins aligned or anti-aligned with the orbital angular momentum. Hughes [46 ] investigated the accuracy with which the redshift can be estimated (if the cosmological parameters are derived independently), and considered the black-hole ring-down phase in addition to the inspiralling signal. Seto [89 ] included the effect of finite armlength (going beyond the long wavelength approximation) and found that the accuracy of the distance determination and angular resolution improve. This happens because the response of the instrument when the armlength is finite depends strongly on the location of the source, which is tightly correlated with the distance and the direction of the orbital angular momentum. Vecchio [97 ] provided the first estimate of parameters for precessing binaries when only one of the two supermassive black holes carries spin. He showed that modulational effects decorrelate the binary parameters to some extent, resulting in a better estimation of the parameters compared to the case when spins are aligned or antialigned with orbital angular momentum. Hughes and Menou [47 ] studied a class of binaries, which they called “golden binaries,” for which the inspiral and ring-down phases could be observed with good enough precision to carry out valuable tests of strong-field gravity. Berti, Buonanno and Will [21 ] have shown that inclusion of non-precessing spin-orbit and spin-spin terms in the gravitational-wave phasing generally reduces the accuracy with which the parameters of the binary can be estimated. This is not surprising, since the parameters are highly correlated, and adding parameters effectively dilutes the available information. Update