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.
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