  The duration of quasar accretion episodes is a key quantity for
distinguishing between models for the formation and growth of supermassive
black holes, the evolution of quasars, and their potential feedback effects on
their host galaxies. However, this critical timescale, often referred to as the
quasar lifetime, is still uncertain by orders of magnitude ($t_{\rm Q} \simeq
0.01\,{\rm Myr} - 1\,{\rm Gyr}$). Absorption spectra of quasars exhibiting
transmission in the He II Ly$\alpha$ forest provide a unique opportunity to
make precise measurements of the quasar lifetime. Indeed, the size of a
quasar's He II proximity zone, the region near the quasar itself where its own
radiation dramatically alters the ionization state of the surrounding
intergalactic medium (IGM), depends sensitively on its lifetime for $t_{\rm
Q}\lesssim 30$ Myr, comparable to the expected $e$-folding time-scale for SMBH
growth $t_{\rm S}=45$ Myr. In this study we compare the sizes of He II
proximity zones in the Hubble Space Telescope (HST) spectra of 6 z~4 quasars to
theoretical models generated by post-processing cosmological hydrodynamical
simulations with $1$D radiative transfer algorithm. We introduce a Bayesian
statistical method to infer the lifetimes of individual quasars which allows us
to fully marginalize over the unknown ionization state of the surrounding IGM.
We measure lifetimes $0.63^{+0.82}_{-0.40}$ Myr and $5.75^{+4.72}_{-2.74}$ Myr
for two objects. For the other 4 quasars large redshift uncertainties undermine
our sensitivity allowing us to only place upper or lower limits. However a
joint analysis of these four systems yields a measurement of their average
lifetime of $\langle t_{\rm Q}\rangle = 1.17^{+1.77}_{-0.84}$ Myr. We discuss
our short ~1 Myr inferred lifetimes in the context of other quasar lifetime
constraints and the growth of SMBHs.
