  We investigate critical quantum metrology,that is the estimation of
parameters in many-body systems close to a quantum critical point, through the
lens of Bayesian inference theory. We first derive a no-go result stating that
any non-adaptive measurement strategy will fail to exploit quantum critical
enhancement (i.e. precision beyond the shot-noise limit) for a sufficiently
large number of particles $N$ whenever our prior knowledge is limited. We then
consider different adaptive strategies that can overcome this no-go result, and
illustrate their performance in the estimation of (i) a magnetic field using a
probe of 1D spin Ising chain and (ii) the coupling strength in a Bose-Hubbard
square lattice. Our results show that adaptive strategies with real-time
feedback control can achieve sub-shot noise scaling even with few measurements
and substantial prior uncertainty.
