Distance and luminosity probability distributions derived from parallax and flux with their measurement errors

We use a Bayesian approach to derive the distance probability distribution for one object from its parallax with measurement uncertainty for two spatial distribution priors, a homogeneous spherical distribution and a galactocentric distribution – applicable for radio pulsars – observed from Earth. We investigate the dependence on measurement uncertainty, and show that a parallax measurement can underestimate or overestimate the actual distance, depending on the spatial distribution prior. We derive the probability distributions for distance and luminosity combined – and for each separately when a flux with measurement error for the object is also available – and demonstrate the necessity of and dependence on the luminosity function prior. We apply this to estimate the distance and the radio and gamma-ray luminosities of PSR J0218+4232. The use of realistic priors improves the quality of the estimates for distance and luminosity compared to those based on measurement only. Use of the wrong prior, for example a homogeneous spatial distribution without upper bound, may lead to very incorrect results