  A Bayesian analysis of the world's $p(\gamma,K^+)\Lambda$ data is presented.
We adopt a Regge-plus-resonance framework featuring consistent interactions for
nucleon resonances up to spin $J = 5/2$. The power of the momentum dependence
of the consistent interaction structure rises with the spin of the resonance.
This leads to unphysical structures in the energy dependence of the computed
cross sections when the short-distance physics is cut off with standard
hadronic form factors. A plausible, spin-dependent modification of the hadronic
form factor is proposed which suppresses the unphysical artifacts. Next, we
evaluate all possible combinations of 11 candidate resonances. The best model
is selected from the 2048 model variants by calculating the Bayesian evidence
values against the world's $p(\gamma,K^+)\Lambda$ data. From the proposed
selection of 11 resonances, we find that the following nucleon resonances have
the highest probability of contributing to the reaction: $S_{11}(1535)$,
$S_{11}(1650)$, $F_{15}(1680)$, $P_{13}(1720)$, $D_{13}(1900)$, $P_{13}(1900)$,
$P_{11}(1900)$, and $F_{15}(2000)$.
