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Coryn A. L. Bailer-Jones developed a statistical treatment of estimating stellar Euclidean distances from noisy stellar parallaxes. His 2015 analysis showed why simply taking the reciprocal of a measured parallax can behave poorly when the fractional uncertainty is appreciable.^[ref]

A Bayesian inference combines the parallax likelihood with a physically chosen distance prior. A positive-distance prior can assign sensible posterior probabilities even to negative measured parallaxes; a negative noisy measurement does not imply a negative physical Euclidean distance.

For a measured stellar parallax $\hat{\varpi }$ with known standard deviation ${\sigma }_{\varpi }>0$, a simple astrometric model is $\hat{\varpi }\mid r\sim N(1/r,{\sigma }_{\varpi }^2)$, using compatible angular and distance units. ThusThis likelihood approaches a positive constant as $r\to \infty$. An unbounded constant-volume-density prior proportional to $r^2$ therefore gives an improper posterior. One proper alternative iswhich combines spherical volume growth with an exponentially decreasing spatial number density. The normalization follows from ${\int }_0^{\infty }{r}^2{e}^{-r/L}dr=2L^3$. Multiplying by the bounded likelihood gives a normalizable posterior even for non-positive measured parallaxes.^[ref]

A negative measured stellar parallax is compatible with a positive true distance under this noisy model; it should not be inverted into a negative distance. When the likelihood becomes nearly flat over the relevant distances, the posterior approaches the chosen prior, whose mode is $2L$. The scale $L$ represents population and survey knowledge, not an additional distance measurement. Testing sensitivity to that choice is part of honest low-signal Bayesian inference.