Costless correction of chain based nested sampling parameter estimation in gravitational wave data and beyond

Nested sampling parameter estimation differs from evidence estimation, in that it incurs an additional source of uncertainty. This uncertainty affects estimates of parameter means and credible intervals in gravitational wave analyses and beyond, and yet, it is typically not accounted for in standard uncertainty estimation methods. In this paper, we present two novel methods to quantify this uncertainty more accurately for any chain based nested sampler, using the additional likelihood calls made at runtime in producing independent samples. Using injected signals of black hole binary coalescences as an example, we first show concretely that the usual uncertainty estimation method is insufficient to capture the true error bar on parameter estimates. We then demonstrate how the extra points in the chains of chain based samplers may be carefully utilised to estimate this uncertainty correctly, and provide a way to check the accuracy of the resulting error bars. Finally, we discuss how this uncertainty affects $p$-$p$ plots and coverage assessments.