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
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