My question is about notation, and whether I'm justified in saying it's confusing (and if I'm right, then what would be a better notation to avoid confusion?)
In "Probabilistic Machine Learning: An Introduction", in section "10.4.2 Bi-tempered loss", the author defines the "tempered version of the log function" as:
And then writes:
This is monotonically increasing and concave, and reduces to the standard (natural) logarithm when $t = 1$.
The author takes this definition from the paper titled "Robust Bi-Tempered Logistic Loss Based on Bregman Divergences", whose authors make the similar statement for the same tempered logarithm function:
The standard (natural) logarithm is recovered at the limit $t \to 1$.
My confusion isn't about the difference between the book's "$t = 1$" and paper's "$t \to 1$" (though I find the latter much easier to understand / 'more correct'), or "reduced versus recovered". My real confusion arises from the notation used on the left-hand side of the definition: I mean the usage of $t$ as subscript in $log_t(x)$, because, if I plot the tempered logarithm function as defined above for various values of the parameter $t$, I see the following:
For $t = 0.4$:
For $t = 0.999999$:
And I can see that these indeed indicate that as "$t \to 1$", the defined function "converges" to the natural logarithm:
So, what's my problem? My problem is that I'm used to perceiving the subscript immediately following the $log$ as the base of the logarithm. In other words, when I see $log_t$, I take $t$ as the base of the logarithm, but obviously that's not the case here, because obviously, $log_{0.999999}(x)$ is a very different function:
Long story short, am I justified in being confused with respect to notation in this example? To me, this looks like authors being sort of "fast and loose" with the notation.
You might say "they clearly indicated that they defined the function as such, so you should not think about your previous experience with the notation used for indicating the base of logarithm function". To which I'd reply as: is there some kind of notation that would avoid this kind of confusion?