“Natural” can perhaps be defined with respect to the physical laws of this universe. So if you sum up the distributions from humans, Turning machines, tape recorders and everything else (properly weighted), you get a distribution for X if you knew X was a number being pulled out this physical universe. A god-distribution, if you wanna call it that—I’m sure you can come up with a better name for it.
Such as ‘universal prior’?
It’s unclear from the initial term-seeking and this response, so just to make sure: it’s a standard term, the usual reference is
M Li, P Vitányi, An Introduction to Kolmogorov Complexity and Its Applications.