An astronomer walks into the office of a mathematician. He says “I’ve come up with an empirical formula that predicts the number of stars at a given distance from the center of the galaxy. Could you take a look at this and tell me what you think?” The mathematician looks at it and says, “No, this can’t possibly be right”. Discouraged, the astronomer leaves.
A week later, the astronomer returns and says “I’ve spent days and nights on this, and I think I have a better formula now. Please take a look.” The mathematician looks at his work and says, “Sorry, this is also wrong.” Even more dejected, the astronomer leaves.
A week after that, the astronomer returns again. He says “I’ve been working on this day and night. I’ve barely slept all week. But I think this new formula must be right.” The mathematician looks at it and sighs. “Look,” he says, “all these formulas you’re showing me don’t even make sense except in the trivial case where all the variables are positive real numbers.”
Eh, the mathematician is maybe taking it a bit far, but I’m often annoyed by empirical formulas that don’t make sense in the context of any sort of model for what they’re supposed to be for. Like, say, if you’re modeling the population of a town, your model ought to be of a functional form that doesn’t allow for negative numbers, even outside of the domain of interest.
On another note, true story, here: I used to teach a college physics lab. One of the labs was on electromagnetic induction, and one of the questions in the lab report was “Why are there no DC transformers?”.
One student answered Because Marvel got the comic book rights.
Not a joke, but a nerdy digression, is that allowed?
A recent change in my field is to allow some model terms to go negative, even though that doesn’t make any “real world” sense. The problem was that by forcing things to be positive, often by squaring terms, a bias was introduced. A significant negative variance component may not be interpretable directly, but it can be taken to mean there is a problem or that the model doesn’t fit the data. In other cases the point value estimate is negative, but the confidence interval includes zero. Forcing point estimate to zero biases the other estimates.
Well, you have to be careful with such things. Sometimes, something going negative that can’t be negative really does have some real-world meaning (for instance, a “negative eccentricity” of an orbit means that you’ve mislabeled which side is the apogee and which is the perigee). And in your situation, it sounds like maybe things need to be negative to make Gaussian errors work, but that might just mean that a Gaussian model for the errors is the wrong model (Gaussians get used all over the place in models, even though they’re wrong more often than statisticians like to admit).
When an eel climbs a ramp
To take squid from a clamp
That’s a moray
Those lines actually appeared in a paper by some biologists who were studying how a moray eel could climb a ramp out of water and they clamped some squid to the top of a ramp and photographed the result.
