Information divergences and the curious case of the binary alphabet

Four problems related to information divergence measures defined on finite alphabets are considered. In three of the cases we consider, we illustrate a contrast which arises between the binary-alphabet and larger-alphabet settings. This is surprising in some instances, since characterizations for the larger-alphabet settings do not generalize their binary-alphabet counterparts. For example, we show that f-divergences are not the unique decomposable divergences on binary alphabets that satisfy the data processing inequality, despite contrary claims in the literature.