  We show that in the continuum limit, the average spectrum method (ASM) is
equivalent to maximizing R\'enyi entropies of order $\eta$, of which Shannon
entropy is the special case $\eta=1$. The order of R\'enyi entropy is
determined by the way the spectra are sampled. Our derivation also suggests a
modification of R\'enyi entropy, giving it a non-trivial $\eta\to0$ limit. We
show that the sharper peaks generally obtained in ASM are associated with
entropies of order $\eta<1$. Our work provides a generalization of the maximum
entropy method that enables extracting more structure than the traditional
method.
