  Gaussian process (GP) modulated Cox processes are widely used to model point
patterns. Existing approaches require a mapping (link function) between the
unconstrained GP and the positive intensity function. This commonly yields
solutions that do not have a closed form or that are restricted to specific
covariance functions. We introduce a novel finite approximation of GP-modulated
Cox processes where positiveness conditions can be imposed directly on the GP,
with no restrictions on the covariance function. Our approach can also ensure
other types of inequality constraints (e.g. monotonicity, convexity), resulting
in more versatile models that can be used for other classes of point processes
(e.g. renewal processes). We demonstrate on both synthetic and real-world data
that our framework accurately infers the intensity functions. Where
monotonicity is a feature of the process, our ability to include this in the
inference improves results.
