Lutz-Kelker bias ($p(\varpi )\propto {\varpi }^{-4}$ for a locally uniform spatial population)

The classical parallax-calibration problem combines measurement errors with the greater volume available at larger distances. A locally uniform spatial population has $p(r)\propto r^2$, which transforms under $r=1/\varpi$ to $p(\varpi )\propto {\varpi }^{-4}$ on the valid distance interval. Parallax cuts and the nonlinear conversion to luminosity can bias calibration samples. A correction is not universal: the population model, selection information and inferential target must be specified. Distinguish the repeated-sampling bias of a selected sample from inference for a particular star using a justified prior.^[ref]

Let true parallax $p$ have prior density proportional to $p^{-4}$ on the valid finite population range, and let an observation supply a constant likelihood on $0<p_{-}\le p\le p_{+}$. The normalized posterior is $3p^{-4}/(p_{-}^{-3}-p_{+}^{-3})$. Integrating gives cumulative probability $(p_{-}^{-3}-p^{-3})/(p_{-}^{-3}-p_{+}^{-3})$, hence the displayed median. A fixed half-width equal to half the observed center gives $p_{-}=p_0/2$, $p_{+}=3p_0/2$ and $p_{\mathrm{med}}=p_0(27/112{)}^{1/3}$. Parameter-dependent error widths require their own normalized likelihood and support.