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I am reading this paper. On page 2, it says that "the likelyhood is monotonically related to the average per-symbol log likelyhood." I know what a monotonic function is. But what does 'monotonically related' mean? Is the idea that $f(x)$ and $g(x)$ are monotonically related if $f(a)<g(b)$ for all inputs such that $a<b$?

Note: tagging this one calculus becuase I know about monotonic from this context. Not sure if that is where it belongs.

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  • $\begingroup$ Your guess is correct; perhaps the paper should have been a bit more explicit so as to avoid this confusion. $\endgroup$ May 5, 2014 at 0:19
  • $\begingroup$ I interpret monotonically related to mean either $f(x),g(x)$ consistently increase and decrease together or else one increases always as the other decreases. $\endgroup$
    – hardmath
    May 5, 2014 at 0:30

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That $f$ is a monotonic function would mean that as $x$ gets bigger, so does $f(x)$, which is the same as saying that as $x$ gets smaller, so does $f(x)$.

That $f(x)$ and $g(x)$ are "monotonically related" means that if $f(x)<f(y)$ then $g(x)<g(y)$. It may be that neither $f$ nor $g$ is monotonic as a function of $x$, but they would be monotonic as functions of each other.

For example, neither $x\mapsto x^2$ nor $x\mapsto |x|$ is a monotonic function, but the two are monotonically related to each other.

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