The answer is simpler than you might think. It is the convention that we call the optimization objective function a "cost function" or "loss function" and therefore, we want to minimize them, rather than maximize them, and hence the negative log likelihood is formed, rather than positive likelihood" in your word. Technically both are correct though. Btw, if we do want to maximize sth, usually we call it "utility function" and hence the goal is the maximize them.

The answer is simpler than you might think. It is the convention that we call the optimization objective function a "cost function" or "loss function" and therefore, we want to minimize them, rather than maximize them, and hence the negative log likelihood is formed, rather than positive likelihood in your word. Technically both are correct though. By the way, if we do want to maximize something, usually we call it "utility function" and hence the goal is to maximize them.