  Determinantal point processes (DPPs) are a class of repulsive point
processes, popular for their relative simplicity. They are traditionally
defined via their marginal distributions, but a subset of DPPs called
"L-ensembles" have tractable likelihoods and are thus particularly easy to work
with. Indeed, in many applications, DPPs are more naturally defined based on
the L-ensemble formulation rather than through the marginal kernel. The fact
that not all DPPs are L-ensembles is unfortunate, but there is a unifying
description. We introduce here extended L-ensembles, and show that all DPPs are
extended L-ensembles (and vice-versa). Extended L-ensembles have very simple
likelihood functions, contain L-ensembles and projection DPPs as special cases.
From a theoretical standpoint, they fix some pathologies in the usual formalism
of DPPs, for instance the fact that projection DPPs are not L-ensembles. From a
practical standpoint, they extend the set of kernel functions that may be used
to define DPPs: we show that conditional positive definite kernels are good
candidates for defining DPPs, including DPPs that need no spatial scale
parameter. Finally, extended L-ensembles are based on so-called ``saddle-point
matrices'', and we prove an extension of the Cauchy-Binet theorem for such
matrices that may be of independent interest.
