  We study the problem of non-parametric Bayesian estimation of the intensity
function of a Poisson point process. The observations are $n$ independent
realisations of a Poisson point process on the interval $[0,T]$. We propose two
related approaches. In both approaches we model the intensity function as
piecewise constant on $N$ bins forming a partition of the interval $[0,T]$. In
the first approach the coefficients of the intensity function are assigned
independent gamma priors, leading to a closed form posterior distribution. On
the theoretical side, we prove that as $n\rightarrow\infty,$ the posterior
asymptotically concentrates around the "true", data-generating intensity
function at an optimal rate for $h$-H\"older regular intensity functions ($0 <
h\leq 1$). In the second approach we employ a gamma Markov chain prior on the
coefficients of the intensity function. The posterior distribution is no longer
available in closed form, but inference can be performed using a
straightforward version of the Gibbs sampler. Both approaches scale well with
sample size, but the second is much less sensitive to the choice of $N$.
Practical performance of our methods is first demonstrated via synthetic data
examples. We compare our second method with other existing approaches on the UK
coal mining disasters data. Furthermore, we apply it to the US mass shootings
data and Donald Trump's Twitter data.
