General form for likelihood of Cox process, from Diggle–Moraga–Rowlingson–Taylor

Question

On page 4 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the following on the top right of the page:

$$ℓ^*(Λ,X)=\prod_{i=1}^nΛ(x_i)\{\int_A Λ(x)dx \}^{-n}$$ is the likelihood for an inhomogeneous Poisson process with intensity $Λ(x)$.

Unfortunately they have not provided a source for their claims, and it seems quite different from the other forms for the likelihood of inhomogeneous Poisson processes that I have seen from other sources, such as here. Although the link shows the log-likelihood, it is clear that corresponding likelihood is not consistent with what Diggle et. al have claimed.

My question is, has anyone seen the proof behind the likelihood formula that is being claimed by Diggle-Moraga-Rowlingson-Taylor? If someone could explain the proof or direct me to a relevant source, I would really appreciate it.

Thank you.

Answer 1

Write the inhomogeneous Poisson point process on a Borel subset $A$ of $\mathbb R^d$ as $$\sum_{i=1}^N\delta_{X_i},$$ where $\delta_a$ is the Dirac probability measure at point $a$, $N$ is the random variable whose value is the number of points that appeared, and the $X_i$'s are the random locations of those points. Then, for each natural $n$, the conditional density $p_n$ of $(X_1,\dots,X_n)$ given that $N=n$ is given by the formula $$p_n(x_1,\dots,x_n)=\Big(\int_A\Lambda(x)\,dx\Big)^{-n}\,\prod_{i=1}^n \Lambda(x_i)\tag1$$ for $(x_1,\dots,x_n)\in A$, provided that $\int_A\Lambda(x)\,dx>0$. So, your likelihood function is this conditional density function $p_n$.

Formula (1) is intuitively obvious, because, at least when $\Lambda$ is continuous and $A$ open, the probability that there will be a point of the Poisson point process in a small neighborhood $U$ of a point $x\in A$ will be approximately $\Lambda(x)|U|$, where $|U|$ is the Lebesgue measure of $U$.

The normalizing factor $\Big(\int_A\Lambda(x)\,dx\Big)^{-n}$ is what makes $p_n$ a probability density. However, for the usual inferences involving the likelihood function (say in the maximum likelihood estimation), the choice of a normalizing factor does not matter.

Up to such an inessential normalizing factor, the answer you linked (given for the case when the dimension is $d=1$) is the same as your likelihood. However, the phrase "distribution of the $n$ first events", used in that answer, does not make sense to me: distribution of [...] events?