As advertised in David Hogg’s recent tweets, Hogg and I re-derived a Lutz-Kelker correction for low SNR parallax during one of our recent group meetings (now held at the Simons Center for Computational Astronomy in NYC). I am including the derivation here for my records. The idea is the following: standard parallax measurements (estimates and their Gaussian errors) can be improved by including prior information. Specifically, the prior for distances in 3D space is \(p(d) = d^2\), and we aim to compute a maximum a posteriori estimate of the parallax given the initial estimate, its error, and the prior.

The full posterior distribution given the parallax estimate and its error is

\[p(\varpi|\hat{\varpi}, \sigma_{\hat{\varpi}}) = p(\hat{\varpi}, \sigma_{\hat{\varpi}} | \varpi) p(\varpi)\]

Finding the maximum of this distribution with a uniform prior would give us the initial estimate. But let’s use the improved parallax prior

\[p(\varpi) = p(d) |\frac{\partial \varpi}{\partial d}| = \varpi^{-4}\]

In this case, taking equating the derivate of the posterior distribution leads to the maximum (a posteriori) estimate

\[\hat{\varpi}^\mathrm{MAP} = \hat{\varpi} \left(\frac{1}{2} + \frac{1}{2}\sqrt{ 1-\left(4\frac{\hat{\varpi}}{\sigma_{\hat{\varpi}}}\right)^2 } \right) = \hat{\varpi} \left(\frac{1}{2} + \frac{1}{2}\sqrt{ 1-16\ \mathrm{SNR}^2 } \right)\]