  Laser frequency noise (LFN) is the dominant source of noise expected in the
Laser Interferometer Space Antenna (LISA) mission, at $\sim$7 orders of
magnitude greater than the typical signal expected from gravitational waves
(GWs). Time-delay interferometry (TDI) suppresses LFN to an acceptable level by
linearly combining measurements from individual spacecraft delayed by durations
that correspond to their relative separations. Knowledge of the delay durations
is crucial for TDI effectiveness. The work reported here extends upon previous
studies using data-driven methods for inferring the delays during the
post-processing of raw phasemeter data, also known as TDI ranging (TDIR). Our
TDIR analysis uses Bayesian methods designed to ultimately be included in the
LISA data model as part of a "Global Fit" analysis pipeline. Including TDIR as
part of the Global Fit produces GW inferences which are marginalized over
uncertainty in the spacecraft separations and allows for independent estimation
of the spacecraft orbits. We demonstrate Markov Chain Monte Carlo (MCMC)
inferences of the six time-independent delays required in the rigidly rotating
approximation of the spacecraft configuration (TDI 1.5) using simulated data.
The MCMC uses fractional delay interpolation (FDI) to digitally delay the raw
phase meter data, and we study the sensitivity of the analysis to the filter
length. Varying levels of complexity in the noise covariance matrix are also
examined. Delay estimations are found to result in LFN suppression well below
the level of secondary noises and constraints on the armlengths to
$\mathcal{O}(30)\ {\rm cm}$ over the ${\sim}2.5\ {\rm Gm}$ baseline.
