  Determinantal point processes (DPPs) are point process models that naturally
encode diversity between the points of a given realization, through a positive
definite kernel $K$. DPPs possess desirable properties, such as exact sampling
or analyticity of the moments, but learning the parameters of kernel $K$
through likelihood-based inference is not straightforward. First, the kernel
that appears in the likelihood is not $K$, but another kernel $L$ related to
$K$ through an often intractable spectral decomposition. This issue is
typically bypassed in machine learning by directly parametrizing the kernel
$L$, at the price of some interpretability of the model parameters. We follow
this approach here. Second, the likelihood has an intractable normalizing
constant, which takes the form of a large determinant in the case of a DPP over
a finite set of objects, and the form of a Fredholm determinant in the case of
a DPP over a continuous domain. Our main contribution is to derive bounds on
the likelihood of a DPP, both for finite and continuous domains. Unlike
previous work, our bounds are cheap to evaluate since they do not rely on
approximating the spectrum of a large matrix or an operator. Through usual
arguments, these bounds thus yield cheap variational inference and moderately
expensive exact Markov chain Monte Carlo inference methods for DPPs.
