Does a classifier based on optimal bayes classifier equation classify every new instance the same way?

Question

I'm trying to understand how optimal bayes classifier works and I was wondering if, given that the function we try to maximize when making a new prediction does not depend on the instance we are trying to classify, is it correct to state that the optimal bayes classifier would always predict the most probable class no matter what the input is?

EDIT:

I've been studying the subject on "Machine Learning, Tom Mitchell, McGraw Hill, 1997" where is stated that the prediction for a new instance is the class $v_j$ for which the function $\underset{v_j \in V}{\operatorname{arg max}}\sum_{h_i \in H}{P}(v_j|h_i){P}(h_i|D) $
Where $V$ is the set of all possible classes, $H$ is the space of the hypothesis and $D$ is the train dataset.

Answer 1

What you seem to miss is that each hypothesis is itself a function that maps input to outputs. According to this post, a hypothesis is "an instance or specific candidate model that maps inputs to outputs and can be evaluated and used to make predictions."

Therefore, the $\arg \max$ in the optimization problem above is over different functions mapping inputs to outputs, taking into account the entire training set.

Once the optimal hypothesis within the hypothesis space is found, it has the ability to classify individual instances to their correct classes.