= Top-hat parallax posterior under a uniform-volume prior
{title2=$p_{\rm med}=[2/(p_-^{-3}+p_+^{-3})]^{1/3}$}
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 $p_{\rm med}=p_0(27/112)^{1/3}$. Parameter-dependent error widths require their own normalized likelihood and support.
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