I am just reading a paper (https://www.aclweb.org/anthology/P19-1400.pdf) and I came across this notation that I have never seen before and do not understand. See given below:
What is the meaning of this dot in the conditional probability?
Dot in conditional probability – explanation of notation
conditional probabilitynotationprobability
Related Solutions
In section 2, the paper gives this definition:
$\min_k$ means "among all possible / suitable $k$, the minimum value of ..."
Whatever comes after $\min_k$ is put into the "..." in the above sentence.
So, $\min_k j_k$ means "among all $k$, the minimum value of $j_k$"
In general, there are many notations using this paradigm of $$\operatorname{Operation}_{\text{index}}\text{expression}$$(where $\text{expression}$ uses $\text{index}$ in some way) to mean "take all the different values of $\text{expression}$ for all suitable $\text{index}$, and use $\operatorname{Operation}$ on them. Examples include
- $\bigcup_k I_k$ meaning "take all the different sets $I_k$, and use $\cup$ on them (take their union)
- $\sum_kf(k)$ meaning "take all the different values $f(k)$, and use $\sum$ on them" (for some reason we use $\sum$, instead of a big $+$).
and so on, for different operations like $\max, \bigcap, \prod, \bigoplus$ and $\bigwedge$. One might even argue that $\lim$ works like this, but that would probably be somewhat of a stretch.
Best Answer
The dot is some (innocuous, but possibly confusing) abuse of notation to show what the probability measure used here is.
Specifically, the KL divergence takes two arguments, both probability measures. Here, the second argument is $p_N^\ast$, which is unambiguous; but the first is the (average) conditional measure with pdf $$ x\mapsto \frac{1}{|D|}\sum_{(m,c)\in D}p_N(x\mid m,c,E^+) $$ The $\cdot$ is used to indicate that, in the expression, the first argument "$\frac{1}{|D|}\sum_{(m,c)\in D}p_N(\cdot\mid m,c,E^+)$" is a function of the variable which goes where the $\cdot$ is.