  We investigate Bayesian and frequentist approaches to resonance searches
using a toy model based on an ATLAS search for the Higgs boson in the diphoton
channel. We draw pseudo-data from the background only model and background plus
signal model at multiple luminosities, from $10^{-3}$/fb to $10^7$/fb. We chart
the change in the Bayesian posterior of the background only model and the
global p-value. We find that, as anticipated, the posterior converges to
certainty about the model as luminosity increases. The p-value, on the other
hand, randomly walks between 0 and 1 if the background only model is true, and
otherwise converges to 0. After briefly commenting on the frequentist
properties of the posterior, we make a direct comparison of the significances
obtained in Bayesian and frequentist frameworks. We find that the well-known
look-elsewhere effect reduces local significances by about 1$\sigma$. We
furthermore find that significances from our Bayesian framework are typically
about 1 to 2$\sigma$ smaller than the global significances, though the
reduction depends on the prior, global significance and integrated luminosity.
This suggests that even global significances could significantly overstate the
evidence against the background only model. We checked that this effect --- the
Bayes effect --- was robust with respect to fourteen choices of prior and
investigated the Jeffreys-Lindley paradox for three of them.
