  We present an approximate Bayesian inference approach for estimating the
intensity of an inhomogeneous Poisson process, where the intensity function is
modelled using a Gaussian process (GP) prior via a sigmoid link function.
Augmenting the model using a latent marked Poisson process and P\'olya--Gamma
random variables we obtain a representation of the likelihood which is
conjugate to the GP prior. We estimate the posterior using a variational
free--form mean field optimisation together with the framework of sparse GPs.
Furthermore, as alternative approximation we suggest a sparse Laplace's method
for the posterior, for which an efficient expectation--maximisation algorithm
is derived to find the posterior's mode. Both algorithms compare well against
exact inference obtained by a Markov Chain Monte Carlo sampler and standard
variational Gauss approach solving the same model, while being one order of
magnitude faster. Furthermore, the performance and speed of our method is
competitive with that of another recently proposed Poisson process model based
on a quadratic link function, while not being limited to GPs with squared
exponential kernels and rectangular domains.
