# How to decide the “best” accuracy score for prediction of binary outcome?

Dr Frank Harrell mentioned in his book and BIOS 330 course that

Accuracy score used to drive model building should be a continuous score that utilizes all the information in the data (e.g. Brier score, log likelihood, deviance, mean square error)

I am wondering:

1. In what sense are these scores "continuous"? Is it continuous when we view it as a mapping from a topological space (input dataset) to $\mathbb{R}$? What would then be the topology/metric on the event space?
2. How is Brier score better than "the proportion classified correctly" as an accuracy score, as Brier score is also sensitive to the relative frequencies of the outcome variable? Consider a non-informative model of always predicting 1 with probability 1, the Brier score would be very different if the true prevalence is 0.30 or 0.005. Or maybe I am not understanding the sensitivity here correctly.
3. How do we choose among the continuous scores? We have Brier score, log likelihood, deviance for the binary prediction case. How do we decide which one would give us the "best" model?