  We report on Bayesian parameter estimation of the mass and equatorial radius
of the millisecond pulsar PSR J0030$+$0451, conditional on pulse-profile
modeling of Neutron Star Interior Composition Explorer (NICER) X-ray
spectral-timing event data. We perform relativistic ray-tracing of thermal
emission from hot regions of the pulsar's surface. We assume two distinct hot
regions based on two clear pulsed components in the phase-folded pulse-profile
data; we explore a number of forms (morphologies and topologies) for each hot
region, inferring their parameters in addition to the stellar mass and radius.
For the family of models considered, the evidence (prior predictive probability
of the data) strongly favors a model that permits both hot regions to be
located in the same rotational hemisphere. Models wherein both hot regions are
assumed to be simply-connected circular single-temperature spots, in particular
those where the spots are assumed to be reflection-symmetric with respect to
the stellar origin, are strongly disfavored. For the inferred configuration,
one hot region subtends an angular extent of only a few degrees (in spherical
coordinates with origin at the stellar center) and we are insensitive to other
structural details; the second hot region is far more azimuthally extended in
the form of a narrow arc, thus requiring a larger number of parameters to
describe. The inferred mass $M$ and equatorial radius $R_\mathrm{eq}$ are,
respectively, $1.34_{-0.16}^{+0.15}$ M$_{\odot}$ and $12.71_{-1.19}^{+1.14}$
km, whilst the compactness $GM/R_\mathrm{eq}c^2 = 0.156_{-0.010}^{+0.008}$ is
more tightly constrained; the credible interval bounds reported here are
approximately the $16\%$ and $84\%$ quantiles in marginal posterior mass.
