In the context of likelihood, why is the log-density considered to be more "natural" than the density? Working through some notes and it says that one of the reasons for using the log-likelihood rather than the likelihood is that the "log-likelihood is a the more "natural" and relevant quantity" in comparison to the likelihood.
I know why the log-likelihood is the more relevant quantity(e.g Fisher information) but why is it considered to be more "natural" than the likeihood?
 A: A simple answer is to think about how likelihood ratios behave. It's a ratio, so it's x:y, where x and y are both non-negative numbers. So the ratio can range from 0 to infinite, with a midpoint at 1. That's pretty inconvenient to work with. If you take log(x:y), you end up with a scale running from negative infinity to positive infinity, with a midpoint at 0.
Having a midpoint in the middle of the scale is nice, but that's not all that happened. Notice that each "unit" in a likelihood ratio is of different value. That is, the difference between 0 to 1 is MUCH larger than the difference from 1 to 2. That is, if you can improve your odds from "impossible" to even, that's enormous. Improving from even to a 2:1 ratio is still pretty good, but not as big a deal. Similarly, improving your odds from 1:1 to 2:1 is still very good, but not nearly so meaningful.
If you use log(x:y), the units end up being of equal value. The different between 0 and 1 is the same as from 1 to 2 is the same as from 2 to 3, and so on.
