  Identifying a reasonably small Hilbert space that completely describes an
unknown quantum state is crucial for efficient quantum information processing.
We introduce a general dimension-certification protocol for both discrete and
continuous variables that is fully evidence-based, relying solely on the
experimental data collected and no other unjustified assumptions whatsoever.
Using the Bayesian concept of relative belief, we take the effective dimension
of the state as the smallest one such that the posterior probability is larger
than the prior, as dictated by the data. The posterior probabilities associated
with the relative-belief ratios measure the strength of the evidence provide by
these ratios so that we can assess whether there is weak or strong evidence in
favor or against a particular dimension. Using experimental data from
spectral-temporal and polarimetry measurements, we demonstrate how to correctly
assign Bayesian plausible error bars for the obtained effective dimensions.
This makes relative belief a conservative and easy-to-use model-selection
method for any experiment.
