  We present Bayesian Analysis of Galaxies for Physical Inference and Parameter
EStimation, or BAGPIPES, a new Python tool which can be used to rapidly
generate complex model galaxy spectra and to fit these to arbitrary
combinations of spectroscopic and photometric data using the MultiNest nested
sampling algorithm. We extensively test our ability to recover realistic
star-formation histories (SFHs) by fitting mock observations of quiescent
galaxies from the MUFASA simulation. We then perform a detailed analysis of the
SFHs of a sample of 9289 quiescent galaxies from UltraVISTA with stellar
masses, $M_* > 10^{10}\ \mathrm{M_\odot}$ and redshifts $0.25 < z < 3.75$. The
majority of our sample exhibit SFHs which rise gradually then quench relatively
rapidly, over $1{-}2$ Gyr. This behaviour is consistent with recent
cosmological hydrodynamic simulations, where AGN-driven feedback in the
low-accretion (jet) mode is the dominant quenching mechanism. At $z > 1$ we
also find a class of objects with SFHs which rise and fall very rapidly, with
quenching timescales of $< 1$ Gyr, consistent with quasar-mode AGN feedback.
Finally, at $z < 1$ we find a population with SFHs which quench more slowly
than they rise, over $>3$ Gyr, which we speculate to be the result of
diminishing overall cosmic gas supply. We confirm the mass-accelerated
evolution (downsizing) trend, and a trend towards more rapid quenching at
higher stellar masses. However, our results suggest that the latter is a
natural consequence of mass-accelerated evolution, rather than a change in
quenching physics with stellar mass. We find $61\pm8$ per cent of $z > 1.5$
massive quenched galaxies undergo significant further evolution by $z = 0.5$.
BAGPIPES is available at https://bagpipes.readthedocs.io
