Are these two equivalent forms for the likelihood of a Poisson point process?

Question

I have a Poisson point process in a bounded region $W$. I'm trying to calculate the likelihood of observing a particular set of points within $W$. I'm told that there are two equivalent forms of likelihood:

$$L(\cdot) = \left( \prod_{i=1}^{n} \lambda(x_i) \right) e^{-\int_W \lambda(u) \, du},$$

which is, apparently, "the more general one" (and the one that I was familiar with as being the correct likelihood), and

$$L(\cdot) = \frac{\left( \int_W \lambda(u) du \right)^N e^{-\int_W \lambda(u) du}}{N!} \times \prod_{i=1}^{N} \lambda(u_i),$$

which I have never seen before, but, apparently, accounts for the occurrence of exactly $N$ points. The problem here is that, as I understand it, assuming we're taking the likelihood of some parameter given some set of points, the first form of likelihood already accounts for the probability of the occurrence of exactly $N$ points, no?

Are these two equivalent forms for the likelihood of a Poisson point process? Or is the first one the correct likelihood, and the second one something else (or even just nonsense)?

Answer 1

I would consider only the first one as being the likelihood of a Poisson point process. It is unclear what the second expression is aiming at.

Point processes are complex and so is the formulation of their likelihood. The tricky part is that the number of points is random, so we cannot think of the likelihood as a "usual" $\mathbb{R}^n \to \mathbb{R}$ mapping. This would restrict us to a fixed dimension $n$. Rather it is intuitive to see it as a collection of mappings, one for each $n \in \mathbb{N}$, where $n$ is the number of points materializing. Then the likelihood of a realization is determined by first counting the total number of points and then plugging in the locations of these points into the respective function.

It is common to choose the components of such a likelihood as the Janossy densities $j_n : \mathbb{R}^n \to \mathbb{R}$ (see my reference, chapter 5 and 7 for a detailed definition). Basically, the value $j_n (x_1, \ldots, x_n) $ encodes the likelihood of $n$ points materializing, one of them in each of the locations $x_1, \ldots, x_n$. For a Poisson point process with intensity $\lambda$ the $n$-th Janossy density is given by $$ j_n (x_1, \ldots, x_n) = \left( \prod_{i=1}^{n} \lambda(x_i) \right) e^{-\int_W \lambda(u) \, du}, $$ which is the first expression in your question. Thus it is commonly referred to as the likelihood of a Poisson point process.

Reference: Daley, D. J. and Vere-Jones, D. (2003). An introduction to the theory of point processes. Volume I, Springer, New York.