  Quantum parameter estimation offers solid conceptual grounds for the design
of sensors enjoying quantum advantage. This is realised not only by means of
hardware supporting and exploiting quantum properties, but data analysis has
its impact and relevance, too. In this respect, Bayesian methods have emerged
as an effective and elegant solution, with the perk of incorporating naturally
the availability of a priori information. In this article we present an
evaluation of Bayesian methods for multiple phase estimation, assessed based on
bounds that work beyond the usual limit of large samples assumed in parameter
estimation. Importantly, such methods are applied to experimental data
generated from the output statistics of a three-arm interferometer seeded by
single photons. Our studies provide a blueprint for a more comprehensive data
analysis in quantum metrology.
