Imagine scientists observe a linear correlation between two measurable quantities x and y in some type of phenomenon. They induce an experimental law: y=kx. However, the degree of precision of their measurements was finite. To what extent can we expect that the correlation will still be linear if, say, we double the level of precision of measurements? I'm not talking about expectations with regards to the value of k, which I assume will be updated, but about the linear form of the law. What kind of inference (inductive, Bayesian, inference to the best explanation) is required to justify such expectation?
My first thought was that the linearity of a correlation could be justified by induction: given that the correlation was linear so far, it is rational to expect that it will remain linear for higher degrees of precision (note that induction is faillible, but it's not the problem I'm interested in here). But this approach is problematic, because there are infinitely many functions that approximate a linear correlation (y=k•sin(x) for example), and all of them would be as much justified by induction, including the ones that no more approximate linearity at higher levels of precision.
We have a similar problem using Bayesian inference: there are infinitely many possible functions that could correspond to our observations, and it's not clear why the ones that will continue to approximate linearity with higher precision would have more weight than the others.
The only inference that seems to justify that the correlation will still be linear is an inference to the best explanation: a linear correlation is more simple, more explanatory, hence more likely than another type of correlation. But this type of inference can be criticised (why would nature be simple?).
So it's not clear why we should expect the form of experimental laws to be preserved with higher precision, and yet this kind of expectation seems to me implicit in scientific reasoning: scientists are not expecting experimental laws to break down every time they increase the precision of their measurement (to some extent at least, I don't mean indefinitely). But maybe I'm wrong? Am I missing something and are there ways of justifying this type of expectation in a Bayesian framework for example? Are there resources in the scientific or philosophical literature on this topic?