  We present a Bayesian analysis of the distances to 15,040 Large Magellanic
Cloud (LMC) RR Lyrae stars using $V$- and $I$-band light curves from the
Optical Gravitational Lensing Experiment, in combination with new $z$-band
observations from the Dark Energy Camera. Our median individual RR Lyrae
distance statistical error is 1.89 kpc (fractional distance error of 3.76 per
cent). We present three-dimensional contour plots of the number density of LMC
RR Lyrae stars and measure a distance to the core LMC RR Lyrae centre of
${50.2482\pm0.0546 {\rm(statistical)} \pm0.4628 {\rm(systematic)} {\rm kpc}}$,
equivalently ${\mu_{\rm LMC}=18.5056\pm0.0024 {\rm(statistical)} \pm0.02
{\rm(systematic)}}$. This finding is statistically consistent with and four
times more precise than the canonical value determined by a recent
meta-analysis of 233 separate LMC distance determinations. We also measure a
maximum tilt angle of $11.84^{\circ}\pm0.80^{\circ}$ at a position angle of
$62^\circ$, and report highly precise constraints on the $V$, $I$, and $z$ RR
Lyrae period--magnitude relations. The full dataset of observed mean-flux
magnitudes, derived colour excess ${E(V-I)}$ values, and fitted distances for
the 15,040 RR Lyrae stars produced through this work is made available through
the publication's associated online data.
