In the introduction to their book "Discriminants, resultants and multidimensional determinant", the authors state a very intriguing observation concerning the coefficients of monomials appearing in $A$-discriminants, which are of the form $vol(\sigma_i)^{vol(\sigma_i)}$, for $\sigma_i$ a simplex (page 5):

"The expression $\prod_ivol(\sigma_i)^{vol(\sigma_i)}$ (or, rather, its logarithm $\sum_i vol(\sigma_i) \log (vol(\sigma_i))$) brings to mind the entropy of a probability distribution. It would be interesting to find a "probabilistic" reason for its appearance in discriminants."

Does any know about the status of this observation, or about references discussing it?