what is a measurement?

I had a great conversation with Hans-Walter Rix (MPIA) to start off the new year. As we often do, we veered into epistemology. He asked: If you can use a data-driven model to infer the Eu/Fe abundance ratio of a star, but your precision is higher than the Cramér–Rao bound or Fisher information that we calculate from any consideration of actual Eu lines (from, say, a physical model of stellar photospheres that we believe). Is then this a measurement of Eu/Fe? What if the data-driven model is in fact not directly measuring Eu but instead measuring elements that are highly covariant with Eu, so that the Eu information is nonetheless very good? This is not a theoretical question, we think this is happening in some cases with our data-driven spectroscopic models.

What if our measurements—made with things covariant with Eu, but not Eu directly—do a good job on predictive accuracy? The machine-learning community (or my stereotype of this community) would say that predictive accuracy is all there is: If you predict Eu/Fe well, then you are measuring Eu/Fe. But most (or many, or my stereotypical) astronomers would say that you aren't measuring Eu/Fe unless Eu lines are involved in the measurement, or some physically motivated derivative of the stellar spectrum with respect to Eu.

Or maybe the problem is causal: It isn't a measurement of Eu if the data aren't causally related to Eu?

I'm not sure I agree with the (cartoon) astronomers here, nor the (cartoon) machine-learners. The situation is complicated. After all, you never directly measure anything of importance in astrophysics; every measurement depends on chains of covariances and common causes. For example, when we measure the age of the Universe, we aren't really measuring the age per se, we are measuring a combination of cosmological parameters that assemble into that age. For instance, you couldn't measure the age of the Universe independently of the Hubble Constant and the mass densities. But I also agree that if Eu isn't in the spectrum, it's a bit weird to say that you can measure it.