  We perform the first simultaneous Bayesian parameter inference and optimal
reconstruction of the gravitational lensing of the cosmic microwave background
(CMB), using 100 deg$^2$ of polarization observations from the SPTpol receiver
on the South Pole Telescope. These data reach noise levels as low as 5.8
$\mu$K-arcmin in polarization, which are low enough that the typically used
quadratic estimator (QE) technique for analyzing CMB lensing is significantly
sub-optimal. Conversely, the Bayesian procedure extracts all lensing
information from the data and is optimal at any noise level. We infer the
amplitude of the gravitational lensing potential to be
$A_\phi\,{=}\,0.949\,{\pm}\,0.122$ using the Bayesian pipeline, consistent with
our QE pipeline result, but with 17\% smaller error bars. The Bayesian analysis
also provides a simple way to account for systematic uncertainties, performing
a similar job as frequentist "bias hardening," and reducing the systematic
uncertainty on $A_\phi$ due to polarization calibration from almost half of the
statistical error to effectively zero. Finally, we jointly constrain $A_\phi$
along with $A_{\rm L}$, the amplitude of lensing-like effects on the CMB power
spectra, demonstrating that the Bayesian method can be used to easily infer
parameters both from an optimal lensing reconstruction and from the delensed
CMB, while exactly accounting for the correlation between the two. These
results demonstrate the feasibility of the Bayesian approach on real data, and
pave the way for future analysis of deep CMB polarization measurements with
SPT-3G, Simons Observatory, and CMB-S4, where improvements relative to the QE
can reach 1.5 times tighter constraints on $A_\phi$ and 7 times lower effective
lensing reconstruction noise.
