Why isn’t the Cramer-Rao lower bound invoked more in applied research?

Raghu Parthasarathy writes:

Here’s a question about statistics, or more accurately the uses of statistics, that has puzzled me for many years: Why isn’t the Cramér-Rao lower bound (CRLB) more often invoked?

As I understand it, the CRLB gives us a bound on how good any estimator of some parameter can be. When I first learned about it—not as an undergraduate, not as a graduate student, not even as a postdoc, but as a faculty member—I was amazed. It’s relevant to work I was doing on finding the locations of single molecules in images, which is relevant to things like super resolution microscopy. I’ve inserted a mention of it in the course I’m currently teaching, on image analysis and related stuff. I’ve often wondered why the CRLB isn’t better known, and more importantly: Why isn’t it used more? Looking at all kinds of studies drawing grand conclusions from noisy data, I think to myself, given the noise, shouldn’t we ask how well even in principle we can know some parameter? What’s the best we can do? Of course, it’s true that we don’t necessarily have a good noise model in many of these cases, but surely we can estimate some kind of curvature of a log-likelihood function given the data distribution itself, a bit like determining the curvature of the blurry image of a fluorescent molecule. Is it that no one asks “What’s the CRLB for the estimator of obesity given noisy gut microbiome abundance data,” or am I missing something about how applicable this actually is?

My quick answer is that the lower bound depends on the model, and once you’ve fit the model, you have your inference, so no need for the lower bound anymore.

To put it another way, Raghu writes, “shouldn’t we ask how well even in principle we can know some parameter?” My answer is, yes, we can do this using fake-data simulation. Or, if the statement is, “surely we can estimate some kind of curvature of a log-likelihood function given the data distribution itself,” then my answer is that this is standard practice if you have a unimodal likelihood. It’s not called the “Cramer-Rao lower bound”; it’s just the normal approximation to the maximum likelihood estimate, or Bayesian inference, or the variational approximation, or whatever. The CRLB is implicit in whatever you’re doing, and once you have your standard error or other measure of uncertainty, there’s no need for the theorem. It’s more that knowledge of the theorem affects the sorts of estimates that we use.