Proper distance prior for parallax inference ($p(r)\propto r^2{e}^{-r/L},r>0$)

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.