Likelihood for a log Gaussian Cox process (LGCP)

Question

Suppose I have a log Gaussian Cox process (LGCP) $X$ with log intensity function $\lambda(x)=S(x)$ where $S$ follows a Gaussian process. Since LGCP still falls under the umbrella of inhomogeneous Poisson processes, the likelihood of an observed point process $(x_1,...,x_n)$ should be proportional to the following density

$$ p(X|S) \propto \prod_{i=1}^n \exp[S(x_i)] \exp[-\exp[S(x_i)]] $$

However I've seen an alternative formulation which replaces the double exponent term dependent on $i$ with a constant equal to the average number of events in the study region.

$$ p(X|S) \propto \prod_{i=1}^n \exp[S(x_i)] \exp\left( -\int_\mathcal{D} \exp[S(x)]dx \right) $$

Which one is correct?

Answer 1

Found the answer in 4.168 of "Statistics for Spatio-temporal data" by Cressie and Wikle. The information in $X$ represents both $n$ the number of points observed AND the exact locations of each of those points. So the likelihood is technically that of $[Z(D_x), \{x_1,...,x_{Z(D_x)}\}]$.