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.
