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

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

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