Quantifying and mitigating bias in inference on gravitational wave source populations

When using incorrect or inaccurate signal models to perform parameter estimation on a gravitational wave signal, biased parameter estimates will in general be obtained. For a single event this bias may be consistent with the posterior, but when considering a population of events this bias becomes evident as a sag below the expected diagonal line of the $P\text{−}P$ plot showing the fraction of signals found within a certain significance level versus that significance level. It would be hoped that recently proposed techniques for accounting for model uncertainties in parameter estimation would, to some extent, alleviate this problem. Here we demonstrate that this is indeed the case. We derive an analytic approximation to the $P\text{−}P$ plot obtained when using an incorrect signal model to perform parameter estimation. This approximation is valid in the limit of high signal-to-noise ratio and nearly correct waveform models. We show how the $P\text{−}P$ plot changes if a Gaussian process likelihood that allows for model errors is used to analyze the data. We demonstrate analytically and using numerical simulations that the bias is always reduced in this way. These results provide a way to quantify bias in inference on populations and demonstrate the importance of utilizing methods to mitigate this bias.

Figure 1

$P\text{−}P$ plots for parameter estimation using the three likelihoods $L^{\prime }(\overrightarrow{\lambda })$, $L(\overrightarrow{\lambda })$, and $\mathcal{L}(\overrightarrow{\lambda })$ shown in the three columns, respectively. In each column the top panel shows a $P\text{−}P$ plot whilst the bottom panel shows the “sag”; i.e. the difference between the ideal diagonal line and the actual $P\text{−}P$ plot. In each panel curves drawn as dotted lines correspond to analytic results whilst solid curves are numerical results. The left-hand column shows ideal, unbiased parameter recovery for the exact likelihood. In the center and right-hand columns different color curves correspond to different distributions of $⟨\delta h({\lambda }_0)|{\tilde{\partial }}_{x}H⟩$. The curves in black are for a constant distribution giving a noncentrality $\mathrm{\Lambda }=2$ [$\mathrm{\Lambda }$ defined in Eq. (31)]. The curves in orange are for a zero mean Gaussian distribution with variance 1. The curves in yellow are for a noncentral Gaussian distribution with mean $4/3$ and variance 1. The curves in light-green are for a noncentral, skewed normal distribution (The PDF of a skew Gaussian distribution with location parameter $\mu$, scale parameter $\sigma$ and skew parameter $\alpha$ is given by $[1+\mathrm{erf}(\alpha (x-\mu )/\sqrt{2}\sigma )]\exp (-(x-\mu {)}^2/2{\sigma }^2)$.) with location parameter 1, scale parameter 1, and skew parameter 1. The curves in dark-green are for a Poisson distribution with mean and variance 1. The curves in blue are for a Gamma distribution with shape parameter 1 and scale parameter 1. And finally, the curves in purple are for a correlated random walk distribution with mean Gaussian step size 1. In all cases the number of parameter dimensions is $N=4$, and the number of points used for the numerical simulations was $n={10}^3$. The left-hand panel clearly shows the exact likelihood does not suffer from any bias, as expected. The center panel shows that in all cases the approximate likelihood suffers from a bias. The right-hand column shows that in all cases the marginalized likelihood reduces the bias relative to the approximate likelihood. In the ideal case (shown in orange) of a zero mean Gaussian distribution for $⟨\delta h({\lambda }_0)|{\tilde{\partial }}_{x}H⟩$ the bias is completely removed.