= Proper distance prior for parallax inference
{title2=$p(r)\propto r^2e^{-r/L},\quad r>0$}
For a measured <stellar parallax> $\widehat\varpi$ with known <standard deviation> $\sigma_\varpi>0$, a simple astrometric model is $\widehat\varpi\mid r\sim N(1/r,\sigma_\varpi^2)$, using compatible angular and distance units. Thus
$$
p(\widehat\varpi\mid r)=\frac1{\sqrt{2\pi}\sigma_\varpi}
\exp\left[-\frac{(\widehat\varpi-1/r)^2}{2\sigma_\varpi^2}\right],\qquad r>0.
$$
This <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 is
$$
p(r)=\frac{r^2}{2L^3}e^{-r/L}\mathbf1_{\{r>0\}},\qquad L>0,
$$
which combines spherical volume growth with an exponentially decreasing spatial <number density>. The <normalization> follows from $\int_0^\infty r^2e^{-r/L}dr=2L^3$. Multiplying by the bounded <likelihood> gives a normalizable <posterior> even for non-positive measured parallaxes.<Estimating distances from parallaxes by Coryn A. L. Bailer-Jones>[Section 7, equations (17)–(19)]{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>.
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