The Lutz-Kelker Paradox
  The Lutz-Kelker correction is intended to give an unbiased estimate for
stellar parallaxes and magnitudes, but it is shown explicitly that it does not.
This paradox results from the application of an argument about sample
statistics to the treatment of individual stars, and involves the erroneous use
of a frequency distribution in the manner of a probability density function
considered as a Bayesian prior. It is shown that the Bayesian probability
distribution for true parallax given the observed parallax of a selected star
is independent of the distribution of other stars. Consequently the Lutz-Kelker
correction should not be used for individual stars. This result has important
implications for the RR Lyrae scale and for the interpretation of results from
Gaia and Hipparcos. The Lutz-Kelker correction is a poor treatment of the
Trumpler-Weaver bias which affects parallax limited samples. A true correction
is calculated using numerical integration and confirmed by a Monte Carlo
method.
