  In a regime where data are only mildly informative, prior choices can play a
significant role in Bayesian statistical inference, potentially affecting the
inferred physics. We show this is indeed the case for some of the parameters
inferred from current gravitational-wave measurements of binary black hole
coalescences. We reanalyze the first detections performed by the twin LIGO
interferometers using alternative (and astrophysically motivated) prior
assumptions. We find different prior distributions can introduce deviations in
the resulting posteriors that impact the physical interpretation of these
systems. For instance, (i) limits on the $90\%$ credible interval on the
effective black hole spin $\chi_{\rm eff}$ are subject to variations of $\sim
10\%$ if a prior with black hole spins mostly aligned to the binary's angular
momentum is considered instead of the standard choice of isotropic spin
directions, and (ii) under priors motivated by the initial stellar mass
function, we infer tighter constraints on the black hole masses, and in
particular, we find no support for any of the inferred masses within the
putative mass gap $M \lesssim 5 M_\odot$.
