  We present a comprehensive study of spectroscopic radius measurements of
twelve neutron stars obtained during thermonuclear bursts or in quiescence. We
incorporate, for the first time, a large number of systematic uncertainties in
the measurement of the apparent angular sizes, Eddington fluxes, and distances,
in the composition of the interstellar medium, and in the flux calibration of
X-ray detectors. We also take into account the results of recent theoretical
calculations of rotational effects on neutron star radii, of atmospheric
effects on surface spectra, and of relativistic corrections to the Eddington
critical flux. We employ Bayesian statistical frameworks to obtain neutron star
radii from the spectroscopic measurements as well as to infer the equation of
state from the radius measurements. Combining these with the results of
experiments in the vicinity of nuclear saturation density and the observations
of ~2 Msun neutron stars, we place strong and quantitative constraints on the
properties of the equation of state between approximately 2-8 times the nuclear
saturation density. We find that around M=1.5 Msun, the preferred equation of
state predicts radii between 10.1 - 11.1 km. When interpreting the pressure
constraints in the context of high density equations of state based on
interacting nucleons, our results suggest a relatively weak contribution of the
three-body interaction potential.
