Aggregate for Entropy over Cohort

Question

Assume I have a black-box predictor that behaves stochastically and I want to capture its uncertainty wrt categorical output $c_j \in \{c_1, c_2, ..., c_J\}$ when given the same input multiple times. A useful metric is the entropy with respect to a single sample: $E(x) = - \sum_j P(c_j|x)\log (c_j|x)$. Now, I am looking for a useful aggregate that reflects the entropy for the cohort $X = \{x_1, x_2, ..., x_I\}$. My first idea was to just aggregate it additively $$ E(X) = -\sum_i\sum_j P(c_j|x_i)\log P(c_j|x_i)$$ and normalize it to the range $[0,1]$ where $1/I$ is the uniform probability for a single category $c_i$; $J\cdot \frac{1}{J}$ cancels out; I is cohort size: $$ E_{norm}(X) = E(X) / I\log (1/J) $$ This is basically the mean entropy as if I had computed it sample-wise and took the average. My problem with that is that it might be misleading as mean is a linear aggregate applied to a non-linear metric. Do you think it is sound or do other aggregates reflect more direct the chaos/determinism of a predictor? Aside I also report agreement with ground truth (Cohen's kappa) and accuracy. So, these aspects are covered.