  Motivated primarily by the large uncertainties in the thermonuclear rate of
the $^{30}$P$(p,\gamma)^{31}$S reaction that limit our understanding of
classical novae, we carried out lifetime measurements of $^{31}$S excited
states using the Doppler Shift Lifetimes (DSL) facility at the TRIUMF Isotope
Separator and Accelerator (ISAC-II) facility. The $^{31}$S excited states were
populated by the $^{3}$He$(^{32}$S$,\alpha)^{31}$S reaction. The deexcitation
$\gamma$ rays were detected by a clover-type high-purity germanium detector in
coincidence with the $\alpha$ particles detected by a silicon detector
telescope. We have applied modern Markov chain Monte Carlo-based Bayesian
methods to perform lineshape analyses of Doppler-shift attenuation method
$\gamma$-ray data for the first time. We have determined the lifetimes of the
two lowest-lying $^{31}$S excited states. First experimental upper limits on
the lifetimes of four higher-lying states have been obtained. The experimental
results were compared to shell-model calculations using five universal
$sd$-shell Hamiltonians. Evidence for $\gamma$ rays originating from the
astrophysically important $J^\pi=3/2^+$, 260-keV $^{30}$P$(p,\gamma)^{31}$S
resonance has also been observed, although strong constraints on the lifetime
will require better statistics.
