 Proper distance prior for parallax inference

For a Gaussian measured stellar parallax, the likelihood at distance $r$ approaches a nonzero constant as $r\to \infty$. An unbounded spatial-volume prior proportional to $r^2$ therefore gives an improper posterior. A finite survey/population model or a decaying density is needed. The illustrative proper prior displayed has normalizing constant $1/(2L^3)$; its scale is a model choice requiring justification, not a new measured datum. prior sensitivity is especially important at low parallax signal-to-noise.