Waveform likelihood with waveform model uncertainties¶

This page details the derivation and the behaviour of the matched-filtering likelihood function for heron waveform models.

The approach taken to derive this is closely related to the approach described in [MooreBerryChuaGair16], however we do not take the approach of adding a statistical waveform difference to an existing approximant, and instead use a statistical model which produces the waveform as a statistical distribution.

Given measured data \(d(f)\) which is composed of some signal \(s(f)\) and stationary Gaussian noise \(n(f)\) which has a power spectral density \(S_n(f)\) [MooreColeBerry15], that is

\[d(f) = s(f) + n(f),\]

then we can perform matched filtering to analyse the signal \(s(f)\) using some waveform model \(h(f,\vec{\lambda})\).

For convenience from this point forward we define \(s \gets s(f)\), and \(h(\vec{\lambda}) \gets h(f, \vec{\lambda})\).

From Bayes Theorem

\[p(\vec{\lambda} | s) = \frac{ p'(s | \vec{\lambda}) p(\vec{\lambda}) }{ p(s) }\]

where \(p'(s|\vec{\lambda})\) is the likelihood that the signal \(s\) is a realisation of the model \(h(\lambda)\).

This likelihood is then

\[p'(s | \vec{\lambda}) \propto \exp \left( - \frac{1}{2} \left\langle s - h(\vec{\lambda}) | s - h(\vec{\lambda}) \right\rangle \right)\]

which introduces the noise-weighted inner product of two vectors,

\[\langle x | y \rangle = 4 \mathfrak{R} \left\{ \sum_{\kappa=1}^M \frac{x(f_{\kappa}) y^*(f_\kappa)}{S_n(f_{\kappa})} \delta f \right\}\]

with \(\kappa\) labelling the \(M\) frequency bins witha resolution \(\delta f\).

The models we use for the gravitational waveform are known to be imperfect, however if we imagine a perfect waveform, we can define a likelihood function \(p(s|\vec{\lambda})\) which represents this model. For a good approximate model then \(p(s|\vec{\lambda}) \approx p'(s|\vec{\lambda})\).

The conventional approach to improving this agreement is to seek ever better approximate models. The approach outlined in [MooreBerryChuaGair16] works by modelling the difference between the “true” likelihood and the approximate one using Gaussian process regression.

Here we take a third approach. Each waveform drawn from the Heron model is a draw from a probability distribution; given the probabilistic nature of the waveform it is necessary to include the probability of the waveform in the likelihood function, and then marginalise this out as a nuisance parameter, that is

\[p(s | \vec{\lambda}) \propto \int p(h(\vec{\lambda})) \exp\left(-\frac{1}{2} \langle s-h(\vec{\lambda}) | s-h(\vec{\lambda}) \rangle \right) \mathrm{d}h(\vec{\lambda})\]

The expression for \(p(h(\vec{\lambda})\) for the heron model is analytic, by virtue of it being a Gaussian process. Letting \(h \gets h(\lambda)\),

\[ \begin{align}\begin{aligned}p(h(\vec{\lambda})) &= \frac{1}{ \sqrt{(2 \pi)^{k} |K|} } \exp \left( - \frac{1}{2} (h - \mu)^T K^{-1} (h-\mu) \right)\\&= \frac{1}{ \sqrt{(2 \pi)^{k} |K|} } \exp \left( - \frac{1}{2} \sum_{i,j}^{M} [K^{-1}]_{i,j} (h-\mu)(f_i) (h-\mu)^*(f_j) \right)\\&= \frac{1}{ \sqrt{(2 \pi)^{k} |K|} } \exp \left( - \frac{1}{2} (h-\mu | h-\mu) \right)\end{aligned}\end{align} \]

with \(\mu \gets \mu(\vec{\lambda})\) and \(K \gets K(\vec{\lambda})\) respectively the mean and the covariance matrix of the Gaussian process evaluated at \(\vec{\lambda}\) for a set of frequencies \(f_1 \cdots f_M\). For convenience we can introduce the notation \((x|y)\) for the inner product weighted by the model variance.

The full likelihood expression is then the integral of the product of Gaussians, which is analytical, giving

\[p(s | \vec{\lambda}) \propto \frac{1}{1+ \prod_{\kappa=1}^{M} \sigma^2(f_{\kappa}, \vec{\lambda}) / S_n(f_{\kappa})} \exp\left( - \frac{1}{2} \cdot 4 \mathfrak{R} \left\{ \sum^M_{\kappa=1} \frac{ (s(f_\kappa)-\mu(f_\kappa, \vec{\lambda}))(s(f_\kappa)-\mu(f_\kappa, \vec{\lambda}))^* }{S_n(f_{\kappa}) + \sigma^2(f_\kappa, \vec{\lambda})} \delta f \right\} \right)\]

MooreColeBerry15: C. J. Moore, R. H. Cole, and C. P. L. Berry. Gravitational-wave sensitivity curves. Classical and Quantum Gravity, 32(1):015014, January 2015. arXiv:1408.0740, doi:10.1088/0264-9381/32/1/015014.
MooreBerryChuaGair16(1,2): Christopher J. Moore, Christopher P. L. Berry, Alvin J. K. Chua, and Jonathan R. Gair. Improving gravitational-wave parameter estimation using Gaussian process regression. \prd , 93(6):064001, March 2016. arXiv:1509.04066, doi:10.1103/PhysRevD.93.064001.