Island-size distributions of submonolayer Ag films on Pd(111) adsorbed
at 90 K and after annealing of the film are recovered from two-photon
photoemission spectra of the first image state. The inversion of the
ill-conditioned problem with the maximum-entropy method reveals magic
numbers in the island-size distributions. Hypothesis testing within th
e framework of Bayesian probability theory indicates a critical nucleu
s size i = 1. After annealing of the film large islands coexist with s
mall clusters in a two-phase state.