In my experience I’ve never found an instance where you would use Brier scores over cross entropy/Bernoulli/Binomial log likelihoods. Does anybody know a concrete example when you would prefer Brier??
As Frank Harrell wrote on his blog (https://www.fharrell.com/post/class-damage/), one advantage of the Brier score could be its interpretability and the ability to break it decompose it into discrimination and calibration components.
For example, for logistic regression, things become a lot simpler if one chooses log loss (equivalently KL divergence) because one ends up with a convex minimization problem. Had one chosen Brier score here the problem is no longer convex and where one starts the training iteration will determine where the updates converge to. Sometimes this indeterminacy is a problem -- am getting poor results, is it because the data has changed, or is it that my initial seed has changed and the udates have converged to a worse solution.
Real decision problems contain a lot of nonlinearities if decomposed the wrong way. The only way to decompose it is as a linear combination of probability and utility (because the utility swallows the nonlinearities). But for each component both probability and utility matters in determining the overall value of the decision.