  This paper considers the posterior contraction of non-parametric Bayesian
inference on non-homogeneous Poisson processes. We consider the quality of
inference on a rate function $\lambda$, given non-identically distributed
realisations, whose rates are transformations of $\lambda$. Such data arises
frequently in practice due, for instance, to the challenges of making
observations with limited resources or the effects of weather on detectability
of events. We derive contraction rates for the posterior estimates arising from
the Sigmoidal Gaussian Cox Process and Quadratic Gaussian Cox Process models.
These are popular models where $\lambda$ is modelled as a logistic and
quadratic transformation of a Gaussian Process respectively. Our work extends
beyond existing analyses in several regards. Firstly, we consider
non-identically distributed data, previously unstudied in the Poisson process
setting. Secondly, we consider the Quadratic Gaussian Cox Process model, of
which there was previously little theoretical understanding. Thirdly, we
provide rates on the shrinkage of both the width of balls around the true
$\lambda$ in which the posterior mass is concentrated and on the shrinkage of
posterior mass outside these balls - usually only the former is explicitly
given. Finally, our results hold for certain finite numbers of observations,
rather than only asymptotically, and we relate particular choices of
hyperparameter/prior to these results.
