don't apply the Lutz-Kelker correction!

One great research moment today was Stephen Feeney (Flatiron) walking into my office to ask me about the Lutz–Kelker correction. This is a correction applied to parallax measurements to account for the point that there are far, far more stars at lower parallaxes (larger distances) than there are at smaller parallaxes. Because of (what I think of as being) Jacobian factors, the effect is stronger in parallax than it is in distance. The LK correction corrects for what—in luminosity space—is sometimes called Eddington bias (and often wrongly called Malmquist bias). Feeney's question was: Should he be applying this LK correction in his huge graphical model for the distance ladder? And, implicitly, should the supernova cosmology teams have applied it in their papers?

The short answer is: No. It is almost never appropriate to apply the LK correction to a parallax. The correction converts a likelihood description (the likelihood mode, the data) into a posterior description (the posterior mode) under an improper prior. Leaving aside all the issues with the wrongness of the prior, this correction is bad to make because in any inference using parallaxes, you want the likelihood information from the parallax-measuring experiment. If you use the LK-corrected parallax in your inference, you are multiplying in the LK prior and whatever prior you are using in your own inference, which is inconsistent, and wrong!

I suspect that if we follow this line of argument down, we will discover mistakes in the distance-ladder Hubble-constant projects! For this reason, I insisted that we start writing a short note about this.

Historical note: I have a paper with Ed Turner (Princeton) from the late 90s that I now consider totally wrong, about the flux-measurement equivalent of the Lutz-Kelker correction. It is wrong in part because it uses wrong terminology about likelihood and prior. It is wrong in part because there is literally a typo that makes one of the equations wrong. And it is wrong in part because it (effectively) suggests making a correction that one should (almost) never make!