Top-hat parallax posterior under a uniform-volume prior ($p_{\mathrm{med}}=[2/(p_{-}^{-3}+p_{+}^{-3}){]}^{1/3}$)

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.