This looks like a very nice book! I suspect I will be recommending it as supplemental reading at least the next time I teach statistical physics. (I recommended a *lot* of supplemental texts when I taught that this past spring, because the class wasn’t able to meet in person after March…) It calls to mind a good many “sweet sorrows” — ideas that I found quite engaging and worked with for a longish while but was never able to get into a satisfying shape. For example, I had the notion that in a similarity-weighted diversity index

$D_2^Z(p) := \left(\sum_{i j} Z_{i j} p_i p_j\right)^{-1} ,$

the sum over $i$ and $j$ is the expected score in a game whose goal is agreement and the two players play randomly. This leads to the question of what happens when the game involves more than two players — what about contractions of the form $Z_{i j k} p_i p_j p_k$? These arise in quantum information, where the $Z_{i j k}$ are the real parts of certain geometric phases, but we could also run into three-party similarity measures when comparing species. To take an artificial but pretty example, suppose $\{t_1,t_2,\ldots,t_7\}$ are seven different phenotypic traits, that $\{s_1,s_2,\ldots,s_7\}$ are seven species, and that traits and species correspond to points and lines in the Fano plane. Each species has three of the seven traits, every two species have one trait in common, and every trait is found in three species. All pairs are alike; we could say that $Z_{i j} = \frac{1}{3}$ for all $i \neq j$. But not all *triads* are alike, because a set of *three* lines in the Fano plane can either meet at a common point or not. Some sets of three species are more similar than others.

To move a little in the direction of biology, we could take a phylogenetic network. Let $S$ be a set of ancestral species and $T$ be a set of descendant species, with a directed graph $G$ providing paths from $S$ to $T$. These paths might diverge if there is an evolutionary radiation, and they might converge if there is hybridization or horizontal gene transfer. This structure defines a matroid on $T$, whose rank function $r(U)$ for $U \subset T$ is the size of the smallest set of vertices having the property that all paths from $S$ to $U$ must pass through it. (This kind of matroid is known as a gammoid.) The rank function $r$ is a kind of dissimilarity measure for the descendant species. A subset $U \subset T$ is independent in the matroid-theoretic sense if there exists a set of vertex-disjoint paths from $S$ whose ending points are exactly $U$; in biological language, this would say that the species in $U$ do not have a genetic common ancestor. (At least, to find one, you’d have to go back further into the past than $S$.)

The funny thing is that rank functions of matroids behave a lot like Shannon information of sets of random variables. They always satisfy $r(U) \leq r(V)$ for $U \subseteq V$, and they are submodular or strongly subadditive: for all $U, V$,

$r(U \cup V) + r(U \cap V) \leq r(U) + r(V).$

So, we have an entropy-like quantity coming just out of the graph structure, even before we put a probability distribution on the species. I find that a bit odd!

This leads into the topic of higher-order mutual information defined by inclusion-exclusion, as you do in Remark 8.1.11, and how to make sense of it.

## Re: Entropy and Diversity: The Axiomatic Approach

The Great Blue Hole