  The "no-hair" theorem states that astrophysical black holes are fully
characterised by just two numbers: their mass and spin. The gravitational-wave
emission from a perturbed black-hole consists of a superposition of damped
sinusoids, known as \textit{quasi-normal modes}. Quasi-normal modes are
specified by three integers $(\ell,m,n)$: the $(\ell, m)$ integers describe the
angular properties and $(n)$ specifies the (over)tone. If the no-hair theorem
holds, the frequencies and damping times of quasi-normal modes are determined
uniquely by the mass and spin of the black hole, while phases and amplitudes
depend on the particular perturbation. Current tests of the no-hair theorem,
attempt to identify these modes in a semi-agnostic way, without imposing priors
on the source of the perturbation. This is usually known as \textit{black-hole
spectroscopy}. Applying this framework to GW150914, the measurement of the
first overtone led to the confirmation of the theorem to $20\%$ level. We show,
however, that such semi-agnostic tests cannot provide strong evidence in favour
of the no-hair theorem, even for extremely loud signals, given the increasing
number of overtones (and free parameters) needed to fit the data. This can be
solved by imposing prior assumptions on the origin of the perturbed black hole
that can further constrain the explored parameters: in particular, our
knowledge that the ringdown is sourced by a binary black hole merger. Applying
this strategy to GW150914 we find a natural log Bayes factor of $\sim 6.5$ in
favour of the Kerr nature of its remnant, indicating that the hairy object
hypothesis is disfavoured with $<1:600$ with respect to the Kerr black-hole
one.
