Novel Method for Incorporating Model Uncertainties into Gravitational Wave Parameter Estimates

Posterior distributions on parameters computed from experimental data using Bayesian techniques are only as accurate as the models used to construct them. In many applications, these models are incomplete, which both reduces the prospects of detection and leads to a systematic error in the parameter estimates. In the analysis of data from gravitational wave detectors, for example, accurate waveform templates can be computed using numerical methods, but the prohibitive cost of these simulations means this can only be done for a small handful of parameters. In this Letter, a novel method to fold model uncertainties into data analysis is proposed; the waveform uncertainty is analytically marginalized over using with a prior distribution constructed by using Gaussian process regression to interpolate the waveform difference from a small training set of accurate templates. The method is well motivated, easy to implement, and no more computationally expensive than standard techniques. The new method is shown to perform extremely well when applied to a toy problem. While we use the application to gravitational wave data analysis to motivate and illustrate the technique, it can be applied in any context where model uncertainties exist.

Figure 1

The mean waveform difference returned by GPR for four values of the chirp mass. The initial value of the waveform difference is zero and grows as a function of time; this is true independent of the distance in parameter space from points in $\mathcal{D}$.

Figure 2

The GPR error estimate as a function of parameters. At parameter values in $\mathcal{D}$, the error equals zero. The GPR error does not diverge outside of $\mathcal{D}$, but the error tends rapidly to a finite limit of ${\sigma }_f$, which reflects the magnitudes of the waveform differences in the training set.

Figure 3

The panels show the likelihoods $L^{\prime }(\overrightarrow{\lambda })$, $L(\overrightarrow{\lambda })$, and $\mathcal{L}(\overrightarrow{\lambda })$. The exact likelihood is peaked at the true value with value 1 there. The approximate likelihood shows a suppressed and shifted peak. The marginalized likelihood (the solid black curve in the right-hand panel) does an excellent job of approximating the exact likelihood. The dotted curve in the right-hand panel shows the marginalized likelihood obtained using the less densely sampled training set, $\mathcal{D}=\{1.984,2.01,2.036,2.062,2.088\}$. In this case, the marginalized likelihood is broader and shorter, but still a good approximation to the exact likelihood.