bias–variance trade-off, for parallax estimation

It was a very successful day today! Christina Eilers (MPIA) and I performed a set of external validations on our spectroscopic parallax project and it passed them well: our parallax estimates are more precise than Gaia for the red-giant stars we care about, and they seem to be unbiased when we look at the positions of stellar clusters (open and globular). A fight broke out with Hans-Walter Rix (MPIA), who doesn't like that our spectroscopic estimates of parallax sometimes go negative! But we are trying to build something that can be used as a likelihood for distance, so we want it to have the same kind of unbiased properties that the Gaia parallaxes have. That's leading to some friction on the team!

Fundamentally the issue is this: Do you want the best distance estimates you can get? Or do you want a likelihood function that can be multiplied into other likelihoods to obtain better distances given everything you know? If you want the former, then you might take on a lot of bias to get lower variance (more precision). If you want the latter, then you want unbiased likelihood components that can be multiplied together.

Another important distinction is this: Do you want to use many stars in concert to do things like measure the rotation curve or a metallicity gradient? Or do you just want to know an individual star's position? If the former, then you want unbiased likelihood functions that you can combine. If the latter, then you want to take on bias to increase precision.

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