  We introduce the framework of Bayesian relative belief that directly
evaluates whether or not the experimental data at hand supports a given
hypothesis regarding a quantum system by directly comparing the prior and
posterior probabilities for the hypothesis. In model-dimension certification
tasks, we show that the relative belief procedure typically chooses Hilbert
spaces that are never smaller in dimension than those selected from optimizing
a broad class of information criteria, including Akaike's criterion. As a
concrete and focused exposition of this powerful evidence-based technique, we
apply the relative belief procedure to an important application: state
reconstruction of imperfect quantum sources. In particular, just by comparing
prior and posterior probabilities based on data, we demonstrate its capability
of tracking multiphoton emissions using (realistically lossy) single-photon
detectors in order to assess the actual quality of photon sources without
making ad hoc assumptions, thereby reliably safeguarding source integrity for
general quantum-information and communication tasks with Bayesian reasoning.
Finally, we discuss how relative belief can be exploited to carry out
parametric model certification and estimate the total dimension of the quantum
state for the combined (measured) physical and interacting external systems
described by the Tavis--Cummings model.
