Notation check, argmax of values in a set

Question

Among C different possible values for each y and m total y's, I want to show the value with maximum number of $y$ having that value.

Let's assume there are $m$ different functions $y_1$,...,$y_m$ each of which can output a value from the set of possible outcomes {1,...,C}. I want to know what outcome is the most common among all the function outputs.

This is the notation that I have so far:

$\hat{y}_i = \underset{c\in\{1,...,C\}}{\mathrm{argmax}}(\{y^{(j)}=c\}_{j=1}^m)$

Is this a correct notation?

Answer 1

What you are looking for is $$ c^{\star}\in\operatorname{argmax}_{c\in\{1,\ldots,C\}}(|\{i\in\{1,\ldots,m\}\colon y_{i}=c\}|). $$ There were two issues in your formulation:

1. $\operatorname{argmax}$ returns a set (what if more than one element maximizes?), so you should use $\in$ instead of $=$.
2. $\operatorname{argmax}$ acts on collections that have a maximum (in your version, you were passing it a set with no clear ordering, let alone a maximum) so we need to take the cardinality $|\cdot|$.

Another issue, that also appeared in your code is that on the left-hand side, you have $i$. Why? Also, why call it $\hat{y}_i$ instead of something like $c^\star$ or $c_{\text{most}}$?

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The above is a bit verbose, so perhaps

$$ c^{\star}\in\operatorname{argmax}_{c}(|\{i\colon y_{i}=c\}|) $$

is best, since the intended meaning is clear.