I'm not sure what you mean by objectivity and subjectivity in this case. With the example of beliefs, you can think of cross entropy as the negative expected v

I'm not sure what you mean by objectivity and subjectivity in this case. With the example of beliefs, you can think of cross entropy as the negative expected value of the log probability you assigned to an outcome, weighted by the true probability of each outcome. If you assign larger log probabilities to more likely outcomes, the cross entropy will be lower.