Benford's law is not odd if you are believer in the Bayesian interpretation of probability theory. The shape of Benford's distribution actually follows the sha

Benford's law is not odd if you are believer in the Bayesian interpretation of probability theory. The shape of Benford's distribution actually follows the shape of a maximum entropy, most uninformative distributions (the “flatest distribution” in an information theoretic way) for a _magnitude_ value (a value that can only be positive). For example: sizes, lengths and volumes. These units can't go in the negative. Alternatively, positions can go in the negative, the ignorance distribution for position is flat horizontal line and numbers representing a position generally do not follow benford's law. It is not that unintuitive. Take street lengths. Assuming that they follow Benford's law (a log prior) simply means that, for any length of street L, if you pick another random street, you are as likely to pick a street within the length range (L/2 to L) than (L to 2*L). From the original street length, to get a street twice as short, you need to subtract much less than you would have to add to get a street twice as long. That is why this distribution is not flat. Or rather it is flat on the multiplication or division operation, not on addition or subtraction. If instead you'd assume that a street X meter longer is as likely as a street X meter shorter you would end up with impossible probabilities. For example, when picking a random street relative to a street of 1 km, a (1 to 3) kilometer street would be as likely as a (-1 to 1) km street??? If you assumed probabilities were equal for all lengths between 0 and infinity that would mean you think there are likely as many streets measuring a tredecillion billion km long as there are street 5 km long. This is simply not how things are sized in the universe. Smaller things are in greater numbers. Log priors are one of these areas where the math predicts the universe logically and the universe is mirrored by the math beautifully iff you do your calculations properly (Using Bayesianity).