Bayesian versus frequentist upper limits
- Rover, Christian et al
- arXiv:1103.2987AEI-2011-014
 | The integrals to be computed for a frequentist and a Bayesian 90\%~upper limit are very different. The Bayesian integral is computed along the vertical amplitude axis, conditioning on the observed detection statistic value~$d^2=d^2_0$. The frequentist integral goes along the horizontal axis of possible realisations of~$d^2$ for any given amplitude. (Example values here: $N=100$, $\Delta_t=1$, $\sigma^2=1$, $k=49$, $d^2_0=11$.) |
 | The mapping from observable~$d^2$ to the upper limit on amplitude. The bottom panel shows the ``background'' distribution of $d^2$ under $H_0$. (Example values here: $N=100$, $\Delta_t=1$, $\sigma^2=1$, $k=49$.) |
 | The distribution of upper limits as a function of amplitude (left panel) and trials factor (for zero amplitude; right panel). Note that the frequentist 90\% limit is essentially a statistic that is designed to have its 10\%~quantile at the true amplitude value. |
 | Illustration of the determination of a 90\%~\textsl{detection sensitivity} threshold. Such a statement would be independent of the observed data, and it requires the specification of an additional parameter: the corresponding false alarm rate defining the threshold of what is considered a ``detection''. (Here: $N=100$, $\Delta_t=1$, $\sigma^2=1$, $k=49$.) |