How to estimate distance and velocity from parallax and proper motion

If the observed parallax ϖ' has a gaussian measurement error σ, there is a 68% probability that the actual parallax ϖ is in the range ϖ' - σ < ϖ < ϖ' + σ (the frequentist approach). The probability distribution within this range is not known from ϖ' and σ alone, and in particular, we cannot state that the most probable distance D is given by D = 1/ϖ'. To obtain a probability distribution, we need to know or assume a distribution of pulsar distances. Similar assumptions are also required to estimate the velocity distribution of radio pulsars.