  Nested sampling parameter estimation differs from evidence estimation, in
that it incurs an additional source of error. This error affects estimates of
parameter means and credible intervals in gravitational wave analyses and
beyond, and yet, it is typically not accounted for in standard error estimation
methods. In this paper, we present two novel methods to quantify this error
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 error 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 error correctly, and provide a way to check the accuracy of the
resulting error bars. Finally, we discuss how this error affects $p$-$p$ plots
and coverage assessments.
